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

$The \space conic \space \\Ax2+Bxy+Cy2+Dx+Ey+F=0 \\ is \space an \space ellipse \space if \space \\ B^2<4AC \\and \\ a \space hyperbola \space if \\ B^2-4AC>0 \\ According \space to \space question \\ S+kS'=4x^2+0xy+(k-9)y^2-4kx+0y-36=0 \\ is \space an \space ellipse \space if \\ 0^2<4(4)(k-9) \\ 0<16k-144 \\ 144<16k \\ k>9 \\\space ......conditions \space on \space k, for \space an \space ellipse\\ Now \\ an \space a \space hyperbola \space if \\ 0^2-4(4)(k-9)>0 \\ 0-16k+144>0 \\ -16k>-144 \\ k<9 \\\space ......conditions \space on \space k, for \space an \space hyperbola\\$