💡First, I need to understand the problem. A super ball is dropped from 150 feet and rebounds one-third of the distance it falls each time it bounces. I need to calculate how many feet the ball will have traveled when it hits the ground the fifth time.To approach this, I'll break down the ball's movement into descents and ascents. Each time the ball hits the ground, it falls a certain distance and then rebounds to a fraction of that distance. Since the ball is dropped initially, the first descent is 150 feet. After each bounce, the ball ascends to one-third of the previous fall distance and then descends the same distance again.I'll list each descent and ascent up to the fifth hit:- **First Descent**: 150 feet- **First Ascent**: 150 * (1/3) = 50 feet- **Second Descent**: 50 feet- **Second Ascent**: 50 * (1/3) = 50/3 feet- **Third Descent**: 50/3 feet- **Third Ascent**: (50/3) * (1/3) = 50/9 feet- **Fourth Descent**: 50/9 feet- **Fourth Ascent**: (50/9) * (1/3) = 50/27 feet- **Fifth Descent**: 50/27 feetNext, I'll calculate the total distance traveled by summing all the descents and ascents:- **Total Descents**: 150 + 50 + 50/3 + 50/9 + 50/27- **Total Ascents**: 50 + 50/3 + 50/9 + 50/27To simplify the calculation, I'll factor out 50 from the fractions:- **Total Descents**: 150 + 50 + 50*(1/3 + 1/9 + 1/27)- **Total Ascents**: 50 + 50*(1/3 + 1/9 + 1/27)Calculating the sum inside the parentheses:1/3 + 1/9 + 1/27 = (9 + 3 + 1)/27 = 13/27Now, substitute back:- **Total Descents**: 150 + 50 + 50*(13/27) = 200 + 650/27- **Total Ascents**: 50 + 50*(13/27) = 50 + 650/27Adding the total descents and ascents:200 + 650/27 + 50 + 650/27 = 250 + 1300/27Convert 1300/27 to a decimal:1300 ÷ 27 ≈ 48.148Finally, add this to 250:250 + 48.148 ≈ 298.148 feetTherefore, the ball will have traveled approximately 298.15 feet when it hits the ground the fifth time.

💡Okay, so I have this problem here: real numbers (x) and (y) satisfy ((x+5)^2 + (y-12)^2 = 14^2), and I need to find the minimum value of (x^2 + y^2). The options are (A) 2, (B) 1, (C) (sqrt{3}), and (D) (sqrt{2}). Hmm, let me think about how to approach this.First, the equation ((x+5)^2 + (y-12)^2 = 14^2) looks like the equation of a circle. Right, the standard form of a circle is ((x - h)^2 + (y - k)^2 = r^2), where ((h, k)) is the center and (r) is the radius. So in this case, the center of the circle is at ((-5, 12)) and the radius is 14. Got that.Now, I need to find the minimum value of (x^2 + y^2). Wait, (x^2 + y^2) is the square of the distance from the origin to the point ((x, y)). So essentially, I'm looking for the point on the circle that is closest to the origin. The minimum value of (x^2 + y^2) will be the square of the shortest distance from the origin to any point on the circle.Okay, so how do I find the shortest distance from the origin to the circle? I remember that the shortest distance from a point to a circle is the distance from the point to the center of the circle minus the radius, but only if the point is outside the circle. If the point is inside the circle, the shortest distance would be zero because the point is already inside.Let me calculate the distance from the origin to the center of the circle. The center is at ((-5, 12)), so using the distance formula: (sqrt{(-5 - 0)^2 + (12 - 0)^2} = sqrt{25 + 144} = sqrt{169} = 13). So the distance from the origin to the center is 13.The radius of the circle is 14, which is larger than 13. That means the origin is inside the circle because the distance from the origin to the center is less than the radius. Wait, hold on, if the origin is inside the circle, then the minimum distance from the origin to the circle would be the radius minus the distance from the origin to the center. So, that would be (14 - 13 = 1). Therefore, the minimum distance is 1, and the minimum value of (x^2 + y^2) would be (1^2 = 1).But let me double-check that. If the origin is inside the circle, then the closest point on the circle to the origin would be along the line connecting the origin to the center of the circle, right? So, moving from the origin towards the center, the closest point on the circle would be in the direction opposite to the center. Wait, no, actually, if the origin is inside the circle, the closest point on the circle would be in the direction from the origin towards the center, but since the origin is inside, the closest point would actually be in the opposite direction. Hmm, maybe I got that mixed up.Let me visualize it. The center is at (-5, 12), which is in the second quadrant. The origin is at (0,0). So the line connecting the origin to the center goes from (0,0) to (-5,12). The circle is centered at (-5,12) with a radius of 14. Since the distance from the origin to the center is 13, which is less than 14, the origin is inside the circle. Therefore, the closest point on the circle to the origin would be in the direction from the origin towards the center, but since the origin is inside, the closest point is actually along the line from the origin through the center, extended out to the circle. Wait, no, that would be the farthest point. The closest point would be in the opposite direction.Wait, maybe I should think of it as the origin being inside the circle, so the closest point on the circle would be in the direction from the origin away from the center. Let me clarify.The distance from the origin to the center is 13. The radius is 14, so the origin is 13 units away from the center, and the circle extends 14 units from the center. Therefore, the closest point on the circle to the origin is 14 - 13 = 1 unit away from the origin in the direction opposite to the center. So, the minimum distance is 1, and hence the minimum value of (x^2 + y^2) is 1.Alternatively, I can think of this using coordinates. Let me parameterize the circle. Let me set (x = -5 + 14 cos theta) and (y = 12 + 14 sin theta), where (theta) is the parameter varying from 0 to (2pi). Then, (x^2 + y^2) becomes:[(-5 + 14 cos theta)^2 + (12 + 14 sin theta)^2]Expanding this:[25 - 140 cos theta + 196 cos^2 theta + 144 + 288 sin theta + 196 sin^2 theta]Simplifying:[25 + 144 + 196 + (-140 cos theta + 288 sin theta) + 196 (cos^2 theta + sin^2 theta)]Since (cos^2 theta + sin^2 theta = 1), this becomes:[25 + 144 + 196 + (-140 cos theta + 288 sin theta) + 196]Wait, that doesn't seem right. Let me recalculate:Wait, 25 + 144 is 169, plus 196 is 365, and then plus another 196? Wait, no, I think I made a mistake in expanding. Let me do it step by step.First, expand ((-5 + 14 cos theta)^2):[25 - 140 cos theta + 196 cos^2 theta]Then, expand ((12 + 14 sin theta)^2):[144 + 288 sin theta + 196 sin^2 theta]So, adding them together:[25 - 140 cos theta + 196 cos^2 theta + 144 + 288 sin theta + 196 sin^2 theta]Combine like terms:[25 + 144 = 169][196 cos^2 theta + 196 sin^2 theta = 196 (cos^2 theta + sin^2 theta) = 196][-140 cos theta + 288 sin theta]So, total:[169 + 196 + (-140 cos theta + 288 sin theta) = 365 + (-140 cos theta + 288 sin theta)]So, (x^2 + y^2 = 365 - 140 cos theta + 288 sin theta). Now, to find the minimum value, I need to find the minimum of this expression. The minimum occurs when (-140 cos theta + 288 sin theta) is minimized.I recall that an expression of the form (A cos theta + B sin theta) can be rewritten as (R cos (theta - phi)), where (R = sqrt{A^2 + B^2}). But in this case, it's (-140 cos theta + 288 sin theta), which can be written as (R cos (theta + phi)) or something similar.Wait, more accurately, (A cos theta + B sin theta = R cos (theta - phi)), where (R = sqrt{A^2 + B^2}) and (tan phi = B/A). So, in this case, (A = -140) and (B = 288). Therefore, (R = sqrt{(-140)^2 + 288^2}).Calculating that:[R = sqrt{19600 + 82944} = sqrt{102544}]Hmm, let me compute that. 102544 divided by 16 is 6409, which is 80^2 + 9^2? Wait, 80^2 is 6400, so 6409 is 80^2 + 3^2, which is 6400 + 9 = 6409. Wait, but 102544 is 16 * 6409, so sqrt(16 * 6409) = 4 * sqrt(6409). Hmm, sqrt(6409) is approximately 80.056, so R is approximately 4 * 80.056 = 320.224. Wait, that seems too large.Wait, maybe I made a mistake. Let me compute 140^2: 140*140=19600. 288^2: 288*288. Let me compute 288*200=57600, 288*88=25344, so total 57600+25344=82944. So, 19600 + 82944 = 102544. So sqrt(102544). Let me see, 320^2=102400, so sqrt(102544) is 320.224 approximately. So R is about 320.224.But wait, that can't be right because the maximum value of (A cos theta + B sin theta) is R, and the minimum is -R. So, in our case, the expression is (-140 cos theta + 288 sin theta), which can be written as (R cos (theta - phi)), but actually, it's (R sin (theta + phi)) or something similar. Wait, maybe I should write it as (R sin (theta + phi)).Wait, regardless, the maximum value of (A cos theta + B sin theta) is (sqrt{A^2 + B^2}), and the minimum is (-sqrt{A^2 + B^2}). So, in our case, the expression is (-140 cos theta + 288 sin theta), so the maximum value is (sqrt{(-140)^2 + 288^2} = sqrt{19600 + 82944} = sqrt{102544} = 320.224), and the minimum is -320.224.So, going back to (x^2 + y^2 = 365 - 140 cos theta + 288 sin theta), which is 365 + (-140 cos theta + 288 sin theta). The minimum value occurs when (-140 cos theta + 288 sin theta) is minimized, which is -320.224. So, the minimum value of (x^2 + y^2) is 365 - 320.224 ≈ 44.776.Wait, that's conflicting with my earlier conclusion that the minimum distance is 1. There must be a mistake here. Let me think again.Wait, perhaps I made a mistake in the parametrization. Let me try another approach. The minimum value of (x^2 + y^2) is the square of the minimum distance from the origin to the circle. Since the origin is inside the circle, the minimum distance is the distance from the origin to the center minus the radius? Wait, no, that would give a negative number. Wait, no, actually, if the origin is inside the circle, the minimum distance from the origin to the circle is the radius minus the distance from the origin to the center.Wait, let me clarify. The distance from the origin to the center is 13, and the radius is 14. So, the origin is inside the circle because 13 < 14. Therefore, the minimum distance from the origin to the circle is 14 - 13 = 1. Hence, the minimum value of (x^2 + y^2) is (1^2 = 1).But then why did the parametrization approach give me a different result? Maybe I messed up the parametrization. Let me check.Wait, when I parameterized the circle as (x = -5 + 14 cos theta) and (y = 12 + 14 sin theta), I assumed that (theta) varies from 0 to (2pi). But perhaps I should have considered the direction from the origin to the center. Let me think.Alternatively, maybe I should use the formula for the distance from a point to a circle. The distance from the origin to the circle is |distance from origin to center - radius|. Since the origin is inside the circle, it's radius - distance from origin to center, which is 14 - 13 = 1. So, the minimum distance is 1, and hence the minimum value of (x^2 + y^2) is 1.Therefore, the answer should be (B) 1.Wait, but earlier when I tried the parametrization, I got a different result. Maybe I made a mistake in the parametrization. Let me try to compute (x^2 + y^2) again.Given (x = -5 + 14 cos theta), (y = 12 + 14 sin theta).So, (x^2 + y^2 = (-5 + 14 cos theta)^2 + (12 + 14 sin theta)^2).Expanding:[25 - 140 cos theta + 196 cos^2 theta + 144 + 288 sin theta + 196 sin^2 theta]Combine like terms:[25 + 144 = 169][196 cos^2 theta + 196 sin^2 theta = 196 (cos^2 theta + sin^2 theta) = 196][-140 cos theta + 288 sin theta]So, total:[169 + 196 + (-140 cos theta + 288 sin theta) = 365 + (-140 cos theta + 288 sin theta)]So, (x^2 + y^2 = 365 - 140 cos theta + 288 sin theta).Now, to find the minimum value, we need to find the minimum of this expression. The term (-140 cos theta + 288 sin theta) can be written as (R sin (theta + phi)) or (R cos (theta + phi)), where (R = sqrt{(-140)^2 + 288^2}).Calculating (R):[R = sqrt{140^2 + 288^2} = sqrt{19600 + 82944} = sqrt{102544} = 320.224]So, the expression (-140 cos theta + 288 sin theta) can be rewritten as (320.224 sin (theta + phi)), where (phi) is some phase shift. The maximum value of this expression is 320.224, and the minimum is -320.224.Therefore, the minimum value of (x^2 + y^2) is (365 - 320.224 = 44.776). Wait, that contradicts my earlier conclusion that the minimum is 1. What's going on here?Ah, I see the mistake now. When I parameterized the circle, I used (x = -5 + 14 cos theta) and (y = 12 + 14 sin theta), but this parameterization is correct only if (theta) is measured from the positive x-axis. However, the direction from the origin to the center is not aligned with the x-axis, so this parameterization doesn't necessarily capture the point closest to the origin.Instead, the closest point on the circle to the origin lies along the line connecting the origin to the center of the circle. Therefore, I should parameterize the point on the circle in the direction opposite to the center from the origin.Let me consider the vector from the origin to the center, which is ((-5, 12)). The unit vector in this direction is (frac{(-5, 12)}{13}). Therefore, the point on the circle closest to the origin would be in the direction opposite to this vector, scaled by the radius.Wait, no, actually, since the origin is inside the circle, the closest point on the circle is in the direction from the origin towards the center, but extended to the circle. Wait, no, that would be the farthest point. The closest point would be in the direction from the origin away from the center.Wait, let me think carefully. The origin is inside the circle. The line connecting the origin to the center is a line from (0,0) to (-5,12). The closest point on the circle to the origin would be in the direction from the origin opposite to the center. So, if the center is at (-5,12), the direction from the origin towards the center is towards the second quadrant. The opposite direction would be towards the fourth quadrant.Therefore, the closest point on the circle to the origin is along the line from the origin through the center, extended beyond the center by the radius. Wait, no, that would be the farthest point. The closest point is in the opposite direction.Wait, perhaps I should use the formula for the distance from a point to a circle. The distance from the origin to the circle is |distance from origin to center - radius|. Since the origin is inside the circle, the minimum distance is radius - distance from origin to center, which is 14 - 13 = 1. Therefore, the minimum value of (x^2 + y^2) is (1^2 = 1).This makes sense because the origin is inside the circle, so the closest point on the circle is just 1 unit away from the origin. Therefore, the minimum value of (x^2 + y^2) is 1.So, despite the parametrization approach giving me a different result, I think the correct answer is 1. Maybe I made a mistake in the parametrization approach by not considering the correct direction. The geometric approach seems more straightforward here.Therefore, the minimum value of (x^2 + y^2) is 1, which corresponds to option (B).

💡Alright, so I have this geometry problem here about triangle ABC and a circle passing through points B and C. The circle intersects AB and AC at points P and Q, and I need to prove that PQ is always parallel to a fixed line. Hmm, okay, let me try to visualize this.First, I imagine triangle ABC with points B and C. There's a circle passing through B and C, so that's circle O1. This circle intersects AB at P and AC at Q. So, depending on where the circle is, P and Q can move along AB and AC. But no matter where the circle is, PQ should always be parallel to some fixed line. Interesting.I think I need to find what that fixed line is. Maybe it's related to BC or something else in the triangle. Let me recall some geometry concepts. If two lines are parallel, their corresponding angles are equal. Maybe I can use some angle relationships here.Since points B, C, P, and Q lie on the same circle, they form a cyclic quadrilateral. In cyclic quadrilaterals, opposite angles are supplementary. So, angle BPC plus angle BQC equals 180 degrees. Hmm, not sure if that helps directly.Wait, maybe I should look at the angles at points P and Q. Since PQ is a chord of the circle, the angles subtended by PQ at points B and C might be related. Or perhaps I can use the power of a point from A with respect to circle O1.Power of a point says that for point A outside the circle, the product of the lengths from A to P and A to B equals the product from A to Q and A to C. So, AP * AB = AQ * AC. That might be useful, but I'm not sure yet.Another thought: if PQ is parallel to a fixed line, maybe that line is BC. If I can show that PQ is parallel to BC, then that would mean PQ is always parallel to BC, which is a fixed line in the triangle. Let me see if that makes sense.To show PQ is parallel to BC, I can use the converse of the basic proportionality theorem (Thales' theorem). If a line divides two sides of a triangle proportionally, then it is parallel to the third side. So, if AP/AB = AQ/AC, then PQ is parallel to BC.From the power of a point, I have AP * AB = AQ * AC. Let's rearrange this: AP/AQ = AC/AB. Hmm, that's not exactly the ratio I need for Thales' theorem. Wait, if I take reciprocals, AQ/AP = AB/AC. Maybe I can manipulate this further.Alternatively, maybe I can use similar triangles. If triangles APQ and ABC are similar, then PQ would be parallel to BC. For similarity, I need corresponding angles to be equal. Let's see.Since B, C, P, Q are concyclic, angle BPQ equals angle BCQ because they subtend the same arc BQ. Similarly, angle BQP equals angle BPC. Hmm, not sure if that directly helps.Wait, let's consider angles at A. If I can show that angle PAQ is equal to angle BAC, but that's already given since P and Q are on AB and AC. Maybe I need to look at another pair of angles.Alternatively, maybe I can use the concept of homothety. If there's a homothety that maps PQ to BC, then they would be parallel. But I'm not too familiar with homothety, so maybe that's not the easiest approach.Let me go back to the power of a point. AP * AB = AQ * AC. Let's denote AP = x and AQ = y. Then, x * AB = y * AC, so x/y = AC/AB. That means AP/AQ = AC/AB. If I can relate this to the sides of the triangle, maybe I can find a ratio that shows PQ is parallel to BC.Wait, if AP/AB = AQ/AC, then by the converse of Thales' theorem, PQ is parallel to BC. But from the power of a point, I have AP/AQ = AC/AB, which is the same as AP/AB = AQ/AC if I rearrange it. So, AP/AB = AQ/AC implies that PQ is parallel to BC.Yes, that seems to work! So, PQ is always parallel to BC, which is a fixed line in triangle ABC. Therefore, PQ is parallel to the fixed line BC.I think that makes sense. Let me just recap:1. Circle O1 passes through B and C, intersecting AB at P and AC at Q.2. By the power of point A with respect to circle O1, AP * AB = AQ * AC.3. This implies AP/AB = AQ/AC.4. By the converse of Thales' theorem, since the line PQ divides AB and AC proportionally, PQ is parallel to BC.5. BC is a fixed line in triangle ABC, so PQ is always parallel to this fixed line.Yeah, that seems solid. I don't see any gaps in the logic. I think I've got it!

💡Okay, so I have this problem here: Given that log base 3 of m plus log base 3 of n equals 4, I need to find the minimum value of m plus n. The options are A: 4√3, B: 4, C: 9, and D: 18. Hmm, let me think about how to approach this.First, I remember that when you add two logarithms with the same base, you can combine them by multiplying the arguments. So, log base 3 of m plus log base 3 of n should be equal to log base 3 of (m times n). Let me write that down:log₃(m) + log₃(n) = log₃(mn)And according to the problem, this equals 4. So,log₃(mn) = 4Now, to get rid of the logarithm, I can rewrite this equation in exponential form. Since log base 3 of something equals 4, that means 3 raised to the power of 4 equals that something. So,mn = 3⁴Calculating 3⁴, that's 3*3*3*3, which is 81. So,mn = 81Alright, so the product of m and n is 81. Now, the question is asking for the minimum value of m + n. Hmm, how do I find the minimum value of the sum of two numbers when their product is fixed?I think this is a classic optimization problem where you can use the AM-GM inequality. AM-GM stands for Arithmetic Mean - Geometric Mean inequality. It states that for any two positive real numbers, the arithmetic mean is always greater than or equal to the geometric mean. In other words,(m + n)/2 ≥ √(mn)Since m and n are positive (because logarithms are only defined for positive numbers), I can apply this inequality here.Plugging in the values I have:(m + n)/2 ≥ √81√81 is 9, so,(m + n)/2 ≥ 9Multiplying both sides by 2 to solve for m + n,m + n ≥ 18So, the minimum value of m + n is 18. But wait, is this actually achievable? The AM-GM inequality tells me that the minimum is 18, but does there exist m and n such that m + n equals 18 and mn equals 81?Let me check. If m + n = 18 and mn = 81, then m and n must be equal because the AM-GM inequality achieves equality when all the numbers are equal. So, if m = n, then m = n = 9. Let me verify:log₃(9) + log₃(9) = log₃(9*9) = log₃(81) = 4Yes, that works because 3⁴ is 81. So, m = n = 9 satisfies the original equation, and m + n = 18 is achievable.Therefore, the minimum value of m + n is indeed 18, which is option D.Wait, let me just make sure I didn't make any mistakes in my reasoning. I converted the sum of logs into the log of the product, which is correct. Then I applied the AM-GM inequality, which is appropriate here because we're dealing with positive numbers and looking for a minimum sum given a fixed product. The calculation for the inequality seems right, and checking the equality condition confirms that m and n can be 9 each. So, yeah, I think I did this correctly.I guess another way to think about it is using calculus. If I let m be a variable and express n in terms of m, then find the minimum of the sum. Let's try that approach to double-check.Given that mn = 81, I can express n as 81/m. Then, m + n becomes m + 81/m. To find the minimum of this function, I can take the derivative with respect to m and set it equal to zero.Let f(m) = m + 81/mf'(m) = 1 - 81/m²Setting f'(m) = 0,1 - 81/m² = 01 = 81/m²Multiplying both sides by m²,m² = 81Taking square roots,m = 9 or m = -9But since m must be positive, m = 9. Then n = 81/m = 9. So, again, m = n = 9, and m + n = 18.This confirms my earlier result. So, whether I use the AM-GM inequality or calculus, I arrive at the same conclusion that the minimum value of m + n is 18.Just to be thorough, let me consider if there are any other possibilities or if I might have missed something. For example, could m and n be different numbers such that their product is 81 and their sum is less than 18? Let's test some numbers.Suppose m = 27 and n = 3. Then, mn = 81, and m + n = 30, which is greater than 18.If m = 81 and n = 1, then m + n = 82, which is way bigger.What if m = 16 and n = 81/16 ≈ 5.0625? Then, m + n ≈ 16 + 5.0625 ≈ 21.0625, still bigger than 18.Alternatively, if m = 4.5 and n = 18, then m + n = 22.5, which is still larger than 18.It seems that no matter what other positive numbers I choose for m and n that multiply to 81, their sum is always greater than or equal to 18. So, 18 is indeed the minimum.I think I've covered all bases here. I used two different methods—AM-GM inequality and calculus—to confirm the result, and I tested some specific values to see if a smaller sum is possible. Everything points to 18 being the minimum value of m + n.**Final Answer**boxed{D}

💡Okay, so I have this equation here: the sum of the first n odd numbers divided by the sum of the first n even numbers equals 125 over 126. Hmm, I need to find the positive integer n that satisfies this. Alright, let's break this down step by step.First, I remember that the sum of the first n odd numbers is a well-known formula. Let me recall... isn't it n squared? Yeah, I think that's right. So, 1 + 3 + 5 + ... + (2n - 1) equals n². That seems familiar from arithmetic series.Now, the denominator is the sum of the first n even numbers. What's that formula? I think it's n times (n + 1). Let me check: 2 + 4 + 6 + ... + 2n. Yeah, that's an arithmetic series where each term increases by 2. The formula for the sum of an arithmetic series is (number of terms) times (first term plus last term) divided by 2. So, here, the number of terms is n, the first term is 2, and the last term is 2n. So, the sum would be n*(2 + 2n)/2, which simplifies to n*(n + 1). Okay, that makes sense.So now, plugging these into the equation, I have n² divided by n(n + 1) equals 125 over 126. Let me write that out:n² / [n(n + 1)] = 125/126Simplify the left side. The n in the numerator and denominator cancels out, so it becomes n / (n + 1). So now the equation is:n / (n + 1) = 125/126Alright, now I need to solve for n. Let's cross-multiply to get rid of the fractions. So, 126n equals 125(n + 1). Let me write that:126n = 125(n + 1)Expanding the right side, that's 125n + 125. So now, subtract 125n from both sides:126n - 125n = 125Which simplifies to:n = 125Wait, that seems straightforward. So, n is 125. Let me just double-check my steps to make sure I didn't make any mistakes.1. Sum of first n odd numbers is n². Correct.2. Sum of first n even numbers is n(n + 1). Correct.3. Plugging into the equation: n² / [n(n + 1)] simplifies to n / (n + 1). Correct.4. Setting that equal to 125/126 and cross-multiplying: 126n = 125n + 125. Subtracting 125n: n = 125. Seems solid.I don't see any errors in my reasoning. So, yeah, n must be 125. Looking at the options, that's option C. So, I think that's the answer.**Final Answer**boxed{C}

💡Okay, so I have this problem where I need to find the value of ( m^2 ) such that the line ( y = mx + 2 ) intersects the ellipse ( x^2 + 9y^2 = 9 ) exactly once. Hmm, that sounds like it's asking for the condition where the line is tangent to the ellipse. I remember that for a line to be tangent to a conic section like an ellipse, the system of equations should have exactly one solution. So, I think I need to solve these two equations together and set the discriminant of the resulting quadratic equation to zero.Let me start by substituting the expression for ( y ) from the line equation into the ellipse equation. So, replacing ( y ) in ( x^2 + 9y^2 = 9 ) with ( mx + 2 ), I get:[x^2 + 9(mx + 2)^2 = 9]Now, I need to expand this equation. Let's do that step by step. First, expand ( (mx + 2)^2 ):[(mx + 2)^2 = m^2x^2 + 4mx + 4]So, substituting back into the ellipse equation:[x^2 + 9(m^2x^2 + 4mx + 4) = 9]Now, distribute the 9 into the terms inside the parentheses:[x^2 + 9m^2x^2 + 36mx + 36 = 9]Next, I'll combine like terms. The ( x^2 ) terms are ( x^2 ) and ( 9m^2x^2 ), so combining those gives:[(1 + 9m^2)x^2]Then, the linear term is ( 36mx ), and the constant terms are 36 and 9. So, subtracting 9 from both sides to set the equation to zero:[(1 + 9m^2)x^2 + 36mx + 27 = 0]Now, this is a quadratic equation in terms of ( x ). For the line to intersect the ellipse exactly once, this quadratic equation must have exactly one solution. That happens when the discriminant is zero. The discriminant ( D ) of a quadratic equation ( ax^2 + bx + c = 0 ) is given by ( D = b^2 - 4ac ). So, let's calculate the discriminant for our equation.Here, ( a = 1 + 9m^2 ), ( b = 36m ), and ( c = 27 ). Plugging these into the discriminant formula:[D = (36m)^2 - 4(1 + 9m^2)(27)]Let me compute each part step by step. First, ( (36m)^2 ) is:[36m times 36m = 1296m^2]Next, compute ( 4(1 + 9m^2)(27) ). Let's first compute ( 4 times 27 ):[4 times 27 = 108]So, now we have:[108(1 + 9m^2) = 108 + 972m^2]Putting it all together, the discriminant ( D ) becomes:[D = 1296m^2 - (108 + 972m^2)]Simplify this expression by distributing the negative sign:[D = 1296m^2 - 108 - 972m^2]Combine like terms. The ( m^2 ) terms are ( 1296m^2 ) and ( -972m^2 ):[1296m^2 - 972m^2 = 324m^2]So, the discriminant simplifies to:[D = 324m^2 - 108]For the quadratic equation to have exactly one solution, the discriminant must be zero:[324m^2 - 108 = 0]Now, solve for ( m^2 ). First, add 108 to both sides:[324m^2 = 108]Then, divide both sides by 324:[m^2 = frac{108}{324}]Simplify the fraction. Both numerator and denominator are divisible by 108:[frac{108}{324} = frac{1}{3}]So, ( m^2 = frac{1}{3} ).Wait, let me double-check my steps to make sure I didn't make any mistakes. Starting from substituting ( y = mx + 2 ) into the ellipse equation, expanding, and simplifying all seem correct. Then, setting the discriminant to zero for tangency condition, computing each part step by step, and simplifying the discriminant also looks right. The final calculation for ( m^2 ) simplifies to ( frac{1}{3} ), which is one of the answer choices.I think this makes sense. The line is tangent to the ellipse when ( m^2 = frac{1}{3} ), so the answer should be option C.