Let C1 be the circle of radius 1 with the 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 that meets the x-axis at a point B. If AB = √5, then the value of r2 is :
Correct Answer: 2
Solution and Explanation
The value of r2 is 2
Radical Axis and Circle Equation
Radical Axis and Circle Equation
Given: Points \( M \) and \( N \) are the midpoints of line segments \( PQ \) and \( ST \) respectively. The line \( MN \) serves as the radical axis connecting two circles. The equation of the line \( MN \) is:
\(8x + 2y - 18 + r_2 = 0\)
Solution:
We are given the equation of the radical axis \( MN \) as:
\(8x + 2y - 18 + r_2 = 0\)
On solving for \( r_2 \), we find:
\(r_2 = 2\)
Thus, the correct answer is \( r_2 = 2 \).
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Concepts Used:
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