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
Consider points M and N as the midpoints of line segments PQ and ST respectively. As a result, the line MN serves as the radical axis connecting two circles.
Equation of MN : 8x+2y-18+r2=0
on solving r2 is 2.
So, the correct answer is 2
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Concepts Used:
Introduction to Three Dimensional Geometry
In mathematics, Geometry is one of the most important topics. The concepts of Geometry are defined with respect to the planes. So, Geometry is divided into three categories based on its dimensions which are one-dimensional geometry, two-dimensional geometry, and three-dimensional geometry.
Direction Cosines and Direction Ratios of Line
Let's consider line ‘L’ is passing through the three-dimensional plane. Now, x,y, and z are the axes of the plane, and α,β, and γ are the three angles the line making with these axes. These are called the plane's direction angles. So, correspondingly, we can very well say that cosα, cosβ, and cosγ are the direction cosines of the given line L.
Read More: Introduction to Three-Dimensional Geometry