Question

The length of the normal (terminated by the major axis) at a point of the ellipse $\frac{x^{2}}{a^{2}}$+$\frac{y^{2}}{b^{2}}$=1 is-

• A. $\sqrt{rr_{1}}$
• B. $\frac{b}{a}$ (r + r₁)
• C. $\frac{b}{a}$|r– r₁|
• D. Independent of r, r₁, where r, r₁ are the focal distance of the point

Answer 1

Answer: (A)

$\sqrt{rr_{1}}$

Answer 2

Let P є (a cos q, b sin q) be point on ellipse

equation of normal at P $\frac{ax}{\cos\theta}$– $\frac{by}{\sin\theta}$= a² – b²

it meets major axis at Q

Q є$\left( \frac{a^{2} - b^{2}}{a}\cos\theta,0 \right)$

So PQ = $\sqrt{\left( a - \frac{a^{2} - b^{2}}{a}\cos\theta \right)^{2} + b^{2}\sin^{2}\theta}$

= $\sqrt{a^{2}\sin^{2}\theta + b^{2}\cos^{2}\theta}$

= PS = a – ae cos q = r

and PSў = a + ae cos q = r₁

rr₁ = a² (1 – e² cos²q)

= a² – (a² – b²) cos² q

= a² sin² q + b² cos² q

Length of normal l (PQ) = $\sqrt{rr_{1}}$