AP EAMCET · Maths · Ellipse

Let $P$ be any point on the ellipse $7 x^{2} + 16 y^{2} = 112, S$ be a focus, $L$ be the corresponding directrix and $P M$ be the perpendicular distance from $P$ to directrix $L$ . Then $\frac{S P}{P M} =$

A $\frac{1}{4}$
B $\frac{1}{2}$
C $\frac{3}{4}$
D $\frac{1}{\sqrt{2}}$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{3}{4}$

Step-by-step Solution

Detailed explanation

We have, the given equation of an ellipse is 7x2+16y2=112. ⇒x216+y27=1 The eccentricity of the ellipse is, e=1-716=916=34 ∴ SPPM=e ⇒SPPM=34

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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Let P be any point on the ellipse 7x2+16y2=112, S be a focus, L be the corresponding directrix and PM be the perpendicular distance from P to directrix L. Then SPPM=

Step-by-step Solution

Let $P$ be any point on the ellipse $7 x^{2} + 16 y^{2} = 112, S$ be a focus, $L$ be the corresponding directrix and $P M$ be the perpendicular distance from $P$ to directrix $L$ . Then $\frac{S P}{P M} =$