Let H: x^2+y^2+4 x y+8 x+8 y+8=0 be a hyperbola. A line L: x+y+1=0 intersects the hyperbola H at two distinct points. If radius of the circle which touches the hyperbola at the points where H meets the line L is R , then find the value of R

Question

Let H: x^2+y^2+4 x y+8 x+8 y+8=0 be a hyperbola. A line L: x+y+1=0 intersects the hyperbola H at two distinct points. If radius of the circle which touches the hyperbola at the points where H meets the line L is R , then find the value of R^2 .

Quantrex Academy · Accepted Answer

Correct answer: 14

Solving H and L we get aligned 2 x^2+2 x-1 & = + & =-1 ...(1) & = -1 2 ...(2) aligned Eqn. of family of circles passing through A and B is S_D+ _L=0 gathered (x- )(x- )+(y+ +1)(y+ +1)+ (x+y+1)=0 x^2+y^2-( + ) x+ +( + +2) y+( +1)( +1)+ (x+y+1)=0 x^2+y^2+x- 1 2 +y- 1 2 +1(x+y+1)=0 gathered Finally of circles through A and B is x^2+y^2+x( +1)+y( +1)+ -1=0 Now, this circle touches at A( - -1) . d y d x ]_A m_ A C =-1 Diff. hyperbola at A aligned & 2 x+2 y d y d x +4 [ x d y d x +y ]+8+8 d y d x =0 & d y d x [y+2 x+4]+x

Let H: x^2+y^2+4 x y+8 x+8 y+8=0 be a hyperbola. A line L: x+y+1=0 intersects the hyperbola H at two distinct points. If radius of the circle which touches the hyperbola at the points where H meets the line L is R , then find the value of R^2 .

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