Question

if the circles x^2 +y^2 + 2x+2y+6=0 AND x^2+y^2+2ky+k=0 intersect orthogonally, then find 'k' value

Answer 1

Let C1, C2 be the centers and r1, r2 be the radii of the circles S = 0, S′ = 0 respectively.
∴ C1 = (–g, –f), C2 = (–g′, –f′)
2 2 2 2
1 2 r g f c, r g f c = + − = + − ′ ′ ′
Let P be point of intersection of the circles.
The two circles cut orthogonally at P=∠C1PC2 =900= 2 2 2 2 2 2 2 C C C P C P (g g ) (f f ) r r 1 2 1 2 1 2 = + = − + − = + ′ ′
2 2 2 2 2 2 2 2 = + − + + − = + − + + + g g 2gg f f 2ff g f c g f c ′ ′ ′ ′ ′ ′ ′
=− + = − + (2gg 2ff ) (c c ) 2gg 2ff c c.
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