Consider a circle C1 x2y24 x2 y alpha5 Let its mirror image

Question

Consider a circle C_1: x^2+y^2-4 x-2 y=alpha-5. Let its mirror image in the line y=2 x+1 be another circle C_2: 5 x^2+5 y^2-10 f x-10 g y+36=0. Let r be the radius of C_2. Then alpha+r is equal to _________.

DivineJEE · Accepted Answer

We have, C_1: x^2+y^2-4 x-2 y=alpha-5 \ C_1:(x-2)^2+(y-1)^5-5=alpha-5 \ C_1:(x-2)^2+(y-1)^2=(sqrt(alpha))^2 So, centre and radius of C_1 are (2,1) and sqrt(alpha) respectively Now, image of (2,1) along the line y=2 x+1 is, (x-2 / 2)=(y-1 / -1)=(-2(4-1+1) / 2^2+(-1)^2) => (x-2 / 2)=(y-1 / -1)=(-8 / 5) \ => x=(-6 / 5) and y=(13 / 5) Now, ((-6 / 5), (13 / 5)) will be the centre of C_2 Therefore f=(6 / 5) and g=(-13 / 5) Now, radius of C_2=r=sqrt(f^2+g^2-(36 / 5)) => r=sqrt((36 / 25)+(169 / 25)-(36 / 5))=1 \ Because r=1 so, alpha=1 \ Therefore alpha+r=1+1=2 Concept : Image of a point (x_1, y_1) w.r.t. a x+b y+c=0 is (x, y), then (x-x_1 / a)=(y-y_1 / b)=(-2(a x_1+b y_1+c) / (a^2+b^2))

Explanation

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