The locus of the midpoints of the portion of the tangents of the ellipse x22+y21=1 intercepted between the coordinate axes is

TS EAMCET · Maths · Ellipse

The locus of the midpoints of the portion of the tangents of the ellipse $\frac{x^{2}}{2} + \frac{y^{2}}{1} = 1$ intercepted between the coordinate axes is

A $\frac{1}{4 x^{2}} + \frac{1}{2 y^{2}} = 1$
B $2 x^{2} + y^{2} = 1$
C $\frac{1}{2 x^{2}} + \frac{1}{4 y^{2}} = 1$
D $x^{2} + 2 y^{2} = 4$

Verified Solution

Answer & Solution

Correct Answer

(C) $\frac{1}{2 x^{2}} + \frac{1}{4 y^{2}} = 1$

Step-by-step Solution

Detailed explanation

Given: ellipse x22+y21=1 We know that equation of tangent in parametric form is given by xacosθ+ybsinθ=1 So here x2cosθ+y1sinθ=1 Let A & B are intersection points of the tangents on the coordinates axes So…

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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