The line x+y=k meets the curve x^2+y^2-2 x-4 y+2=0 at two points A and B . If O is the origin and AOB =90^ , then the value of k(k>1) is

Step-by-step solution

Eqn. of circle is x^2+y^2-2 x-4 y+2=0 ...(i) Homogenising eqn. (i), we get x^2+y^2-2 x 1-4 y 1+2 (1)^2=0 aligned x^2+y^2-2 x ( x+y k )-4 y ( x+y k )+2 ( x+y k )^2=0 & & x+y=k x+y k =1 aligned array r k^2 x^2+k^2 y^2-2 k x^2-2 k x y-4 k x y-4 k y^2+2 x^2 +2 y^2+4 x y=0 array aligned & x^2 (k^2-2 k+2 )+(4-6 k) x y+ (k^2-4 k+2 ) y^2=0 & Given : A O B=90^ & Coefficient x^2+ Coefficient of y^2=0 & k^2-2 k+2+k^2-4 k+2=0 aligned aligned & 2 k^2-6 k+4=0 & k^2-3 k+2=0 & (k-2)(k-1)=0 & k=2,1 & Given, k>1 k=2 . aligned