Question

Consider two points whose ordinates differ by 3, on the ellipse. The tangents at these variable points, intersect at P. find the locus of P.

• A. ${(\frac{x^2}{25}+\frac{y^2}{9})}^2+(\frac{x^2}{25}+\frac{y^2}{9}-1)\frac{4x^2}{25}=0$
• B. ${(\frac{x^2}{25}+\frac{y^2}{9})}^2-(\frac{x^2}{25}+\frac{y^2}{9}+1)\frac{4x^2}{25}=0$
• C. ${(\frac{x^2}{25}+\frac{y^2}{9})}^2+(\frac{x^2}{25}+\frac{y^2}{9}+1)\frac{4x^2}{25}=0$
• D. ${(\frac{x^2}{25}+\frac{y^2}{9})}^2-(\frac{x^2}{25}+\frac{y^2}{9}-1)\frac{4x^2}{25}=0$

Answer 1

The correct option is D

${(\frac{x^2}{25}+\frac{y^2}{9})}^2-(\frac{x^2}{25}+\frac{y^2}{9}-1)\frac{4x^2}{25}=0$

The roots of the above equation will give the ordinates of the contact points of the two tangents from (h, k) to the ellipse. If β₁ and β₂ be the roots of the above equation, then according to the given condition| β₁- β₂|=3

i.e. (β₁+ β₂)²-4 β₁ β₂=9