TS EAMCET · Maths · Ellipse

$P$ is a point on the conic $a^{2} x^{2} + b^{2} y^{2} = a^{2} (a^{2} + b^{2} - y^{2})$ and $S$ is a focus of that conic. $M$ is the foot of the perpendicular from $P$ on to a directrix of that conic nearer to $S$ . If $P M = K S P$ , then $K =$

A $\frac{b}{\sqrt{a^{2} + b^{2}}}$
B $\frac{\sqrt{a^{2} + b^{2}}}{b}$
C $\frac{a}{\sqrt{a^{2} + b^{2}}}$
D $\frac{\sqrt{a^{2} + b^{2}}}{a}$

Verified Solution

Answer & Solution

Correct Answer

(B) $\frac{\sqrt{a^{2} + b^{2}}}{b}$

Step-by-step Solution

Detailed explanation

Given, equation of conic a2x2+b2y2=a2a2+b2-y2 ⇒a2x2+b2y2+a2y2=a2a2+b2 ⇒a2x2a2a2+b2+y2b2+a2a2a2+b2=1 ⇒x2a2+b2+y2a2=1 …i ∴Equation i represents an ellipse whose eccentricity is given by e=1-a2a2+b2=ba2+b2 Also, we know that in a conic,…

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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P is a point on the conic a2x2+b2y2=a2a2+b2-y2 and S is a focus of that conic. M is the foot of the perpendicular from P on to a directrix of that conic nearer to S. If PM=KSP, then K=

Step-by-step Solution

$P$ is a point on the conic $a^{2} x^{2} + b^{2} y^{2} = a^{2} (a^{2} + b^{2} - y^{2})$ and $S$ is a focus of that conic. $M$ is the foot of the perpendicular from $P$ on to a directrix of that conic nearer to $S$ . If $P M = K S P$ , then $K =$