If the conics whose equations are S ≡ x 2 sin 2 θ+2 h x y+y2 cos 2 θ+32 x+16 y+19=0 S prime ≡ x 2 cos 2 θ+2 h prime xy + y 2 sin 2 θ+16 x +32 y +19=0 intersects in four concylic points then

Question

If the conics whose equations are S ≡ x 2 sin 2 θ+2 h x y+y2 cos 2 θ+32 x+16 y+19=0 S prime ≡ x 2 cos 2 θ+2 h prime xy + y 2 sin 2 θ+16 x +32 y +19=0 intersects in four concylic points then (A)  h+h prime=0 (B) h=h prime (C) h+h prime=1 (D) h+h prime=2

Tardigrade Admin · Accepted Answer

S+λ S prime=0 ⇒ x2( sin 2 θ+λ cos 2 θ)+y2( cos 2 θ+λ sin 2 θ)+2 x y(h+λ h prime)+x(32+16 λ)+y(16+32 λ)+19(1+λ)=0 it will represent a circle it sin 2 θ+λ cos 2 θ= cos 2 θ+λ sin 2 θ h+λ h prime=0 λ=1 ∴ h+h prime=0

Q. If the conics whose equations are $S \equiv x^{2} sin^{2} θ + 2 h x y + y^{2} cos^{2} θ + 32 x + 16 y + 19 = 0$ & $S^{'} \equiv x^{2} cos^{2} θ + 2 h^{'} x y + y^{2} sin^{2} θ + 16 x + 32 y + 19 = 0$ intersects in four concylic points then

$h + h^{'} = 0$

$h = h^{'}$

$h + h^{'} = 1$

$h + h^{'} = 2$

Q. If the conics whose equations are S≡x2sin2θ+2hxy+y2cos2θ+32x+16y+19=0 & S′≡x2cos2θ+2h′xy+y2sin2θ+16x+32y+19=0 intersects in four concylic points then

h+h′=0

h=h′

h+h′=1

h+h′=2

S+λS′=0 ⇒x2(sin2θ+λcos2θ)+y2(cos2θ+λsin2θ)+2xy(h+λh′)+x(32+16λ)+y(16+32λ)+19(1+λ)=0 it will represent a circle it sin2θ+λcos2θ=cos2θ+λsin2θ&h+λh′=0 λ=1∴h+h′=0

Q. If the conics whose equations are $S \equiv x^{2} sin^{2} θ + 2 h x y + y^{2} cos^{2} θ + 32 x + 16 y + 19 = 0$ & $S^{'} \equiv x^{2} cos^{2} θ + 2 h^{'} x y + y^{2} sin^{2} θ + 16 x + 32 y + 19 = 0$ intersects in four concylic points then

$h + h^{'} = 0$

$h = h^{'}$

$h + h^{'} = 1$

$h + h^{'} = 2$