TS EAMCET · Maths · Ellipse

A variable tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ makes intercepts on both the axes. The locus of the middle point of the portion of the tangent between the coordinate axes is

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

Verified Solution

Answer & Solution

Correct Answer

(D) $\frac{a^{2}}{x^{2}} + \frac{b^{2}}{y^{2}} = 4$

Step-by-step Solution

Detailed explanation

Given: The equation of ellipse is x2a2+y2b2=1. Equation of tangent is xacosθ+ybsinθ=1. The tangent meets the axes at Aacosθ,0 and B0,bsinθ. Let x1,y1 be the midpoint of AB. acosθ+02, 0+bsinθ2=x1,y1 ⇒acosθ=2x1⇒cosθ=a2x1…

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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A variable tangent to the ellipse x2a2+y2b2=1 makes intercepts on both the axes. The locus of the middle point of the portion of the tangent between the coordinate axes is

Step-by-step Solution

A variable tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ makes intercepts on both the axes. The locus of the middle point of the portion of the tangent between the coordinate axes is