Answer to Question #91740 in Analytic Geometry for Ra

Question

Let S≡4x²−9y²−36=0 and S'≡y²−4x=0 be two conics. Under what conditions on k, will the conic S+kS'= 0 represent: i) an ellipse? ii) a hyperbola?

Answer 1

The conic Ax2+Bxy+Cy2+Dx+Ey+F=0

is an ellipse if B2<4AC and a hyperbola if B2-4AC>0

S+kS'=4x2+0xy+(k-9)y2-4kx+0y-36=0

is an ellipse if

02<4(4)(k-9)

0<16k-144

144<16k

k>9

an a hyperbola if

02-4(4)(k-9)>0

0-16k+144>0

-16k>-144

k<9