Question

Let PQ be a focal chord of the parabola y 2 = 4 x such that it subtends an angle of π/2 at the point (3, 0). Let the line segment PQ be also a focal chord of the ellipse $E:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,a^2>b^2.$ If e is the eccentricity of the ellipse E , then the value of $\frac{1}{e^2}$ is equal to

• A. $1+\sqrt2$
• B. $3+2\sqrt2$
• C. $1+2\sqrt3$
• D. $4+5\sqrt3$

Answer 1

Answer: (B)

$3+2\sqrt2$

Answer 2

The correct option is(B): $3+2\sqrt2$

$∠PQR=\frac{\pi}{2}$

∴ P ≡ (1, 2) & Q(1, –2)

∴ for ellipse

$\frac{1}{a^2}+\frac{4}{b^2}=1$

and ae = 1

⇒ (5 – e 2)e 2 = 1 – e 2

⇒ e 4 – 6 e 2 + 1 = 0

$⇒\frac{1}{e^2}=3+2\sqrt2$