TS EAMCET · Maths · Ellipse

If $O T$ is the semi-minor axis of an ellipse, $A$ and $B$ are its foci and $∠ A T B$ is a right angle, then the eccentricity of that ellipse is

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

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{1}{\sqrt{2}}$

Step-by-step Solution

Detailed explanation

Let ellipse x2a2+y2b2=1 So we have A-ae,0, Bae,0 & T0,b Now, slope of AT=bae and slope of BT=b-ae Given, AT⊥BT So, bae.b-ae=-1 ⇒b2a2=e2 ...i Eccentricity of ellipse e2=1-b2a2 Using Equation i e2=1-e2 ⇒2e2=1 ⇒e=12

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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If OT is the semi-minor axis of an ellipse, A and B are its foci and ∠ATB is a right angle, then the eccentricity of that ellipse is

Step-by-step Solution

If $O T$ is the semi-minor axis of an ellipse, $A$ and $B$ are its foci and $∠ A T B$ is a right angle, then the eccentricity of that ellipse is