Question

If the normals at the points A, B, C, D of an ellipse meet in a point O, then SA .SB .SC . SD is equal to (where S is one of the foci)-

• A. $\frac{b^{2}}{e^{2}}$. SO²
• B. $\frac{b^{3}}{e^{3}}$. SO³
• C. $\frac{2b^{2}}{e^{2}}$. SO²
• D. $\frac{b^{2}}{3e^{2}}$. 2SO²

Answer 1

Answer: (A)

$\frac{b^{2}}{e^{2}}$. SO²

Answer 2

If S be the point (– ae, 0), we have SA = a + ex₁, etc.

Therefore, SA . SB . SC . SD

= (a + ex₁) (a + ex₂) (a + ex₃) (a + ex₄)

= a⁴ + a³e S x₁ + a²e² S x₁x₂ +

ae³ S x₁x₂x₃ + e⁴x₁x₂x₃x₄

= $\frac{b^{2}}{e^{2}}$ {(h + ae)² + k} ; on substitution and

implification $\frac{b^{2}}{e^{2}}$SO² .

Hence (1) is the correct answer.