Circle 4 Question 2

2. The line x=y touches a circle at the point (1,1). If the circle also passes through the point (1,−3), then its radius is

(2019 Main, 10 April I)

(a) 32

(b) 22

(c) 2

(d) 3

Show Answer

Answer:

Correct Answer: 2. (b)

Solution:

  1. Since, the equation of a family of circles touching line L=0 at their point of contact(x1,y1) is (x−x1)2+(y−y1)2+λL=0, where λ∈R.

∴ Equation of circle, touches the x=y at point (1,1) is

(x−1)2+(y−1)2+λ(x−y)=0

⇒x2+y2+(λ−2)x+(−λ−2)y+2=0

∵ Circle (i) passes through point (1,−3).

1+9+(λ−2)+3(λ+2)+2=0⇒4λ+16=0⇒λ=−4

So, equation of circle (i) at λ=−4, is

x2+y2−6x+2y+2=0

Now, radius of the circle =9+1−2=22.

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