Let C1 be the circle of radius 1 with center at the origin. Let C2 be the circle of radius r with center at the point A=(4,1), where 1<r<3. Two distinct common tangents PQ and ST of C1 and C2 are drawn. The tangent PQ touches C1 at P and C2 at Q. The tangent ST touches C1 at S and C2 at T. Midpoints of the line segments PQ and ST are joined to form a line which meets the x-axis at a point B. If AB=5, then the value of r2 is

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Let $C_{1}$ be the circle of radius 1 with center at the origin. Let $C_{2}$ be the circle of radius r with center at the point A=(4,1), where 1<r<3. Two distinct common tangents PQ and ST of C₁ and C₂ are drawn. The tangent PQ touches C₁ at P and C₂ at Q. The tangent ST touches C₁ at S and C₂ at T. Midpoints of the line segments PQ and ST are joined to form a line which meets the x-axis at a point B. If $AB = \sqrt{5}$ , then the value of r² is

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answer is 2.

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Detailed Solution

$Equations of given circles are \begin{aligned} x^{2} + y^{2} = 1 and \\ (x - 4)^{2} + (y - 1)^{2} = r^{2} \end{aligned}$

Equation of line Midpoints of the line segments PQ and ST ia

$: 8 x + 2 y - 18 + r^{2} = 0$

$B is on x -axis \Rightarrow B (\frac{18 - r^{2}}{8}, 0)$

$\begin{aligned} A B = \sqrt{5} \\ \sqrt{{(\frac{18 - r^{2}}{8} - 4)}^{2} + 1} = \sqrt{5} \end{aligned}$

$r^{2} = 2$

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