Question

If the normal at any point P on ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}}$=1 meets the auxillary circle at Q and R such that ŠQOR = 90ŗ where O is centre of ellipse , then –

• A. a⁴ + 2b⁴³ 3a²b²
• B. a⁴ + 2b⁴³ 5a²b² + 2a³b
• C. a⁴ + 2b⁴³ 3a²b² + ab
• D. None

Answer 1

Answer: (B)

a⁴ + 2b⁴³ 5a²b² + 2a³b

Answer 2

Normal at P(acosq, bsinq) is

ax secq – by cosecq = a² – b²

homogenising with auxillary circle

x² + y² = a² + b²

x² + y² = (a² + b²) $\frac{(ax\sec\theta - by\cos ec\theta)^{2}}{(a^{2} - b^{2})^{2}}$

\ for ŠQOR = 90⁰

Coefft. of x² + Cofft. of y² = 0

1 – $\frac{a^{4}\sec^{2}\theta}{(a^{2} - b^{2})^{2}}$ + 1 – $\frac{a^{2}b^{2}\cos ec^{2}\theta}{(a^{2} - b^{2})^{2}}$ = 0

a⁴ – 5a²b² + 2b⁴ = a⁴tan²q + a²b²cot²q

\ AM ³ GM

a⁴ – 5a²b² + 2b⁴ ³ 2a³b69