The locus of the point of intersection of tangents to the ellipse x^2a^2+y^2b^2=1 which meet at right angles is

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NIMCET 2021

The locus of the point of intersection of tangents to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ which meet at right angles is

Answer (Detailed Solution Below)View answer ▾

Answer: Option A — a circle

Detailed Solution

A tangent of slope $m$ is $y = m x \pm a^{2} m^{2} + b^{2}$ . Through $(h, k)$ : $(k - mh)^{2} = a^{2} m^{2} + b^{2}$ , i.e. $(h^{2} - a^{2}) m^{2} - 2 hk m + (k^{2} - b^{2}) = 0$ .

The two tangents are perpendicular when $m_{1} m_{2} = - 1$ : $\frac{k ^{2} - b ^{2}}{h ^{2} - a ^{2}} = - 1$ , so $h^{2} + k^{2} = a^{2} + b^{2}$ .

The locus $x^{2} + y^{2} = a^{2} + b^{2}$ is a circle (the director circle).

✓ Check

For

a = 3

b = 2

the point

(3, 2)

is on

x^{2} + y^{2} = 13

, and the tangents

x = 3

and

y = 2

from it are perpendicular.

⚠ Common trap

m_{1} m_{2} = - 1

gives equal coefficients of

h^{2}

and

k^{2}

, which is a circle.

Quick tip

Director circle of the ellipse: $x^{2} + y^{2} = a^{2} + b^{2}$

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Ellipse — Preparation

This question tests your understanding of Ellipse, a key topic in the Quantitative Aptitude section. The detailed solution above uses a step-by-step approach to show the reasoning, and where available, a shortcut that helps you solve similar questions in under 90 seconds during the actual exam.

An ellipse is the set of points whose distances from two fixed points (the foci) add up to a constant. Its standard form is x^2/a^2 + y^2/b^2 = 1, where a > b, and its eccentricity e satisfies b^2 = a^2(1 - e^2), with 0 < e < 1. The foci are at (ae, 0) and (-ae, 0) and the directrices are x = a/e and x = -a/e. Exam questions test finding the foci, eccentricity, length of the major and minor axes, latus rectum (2b^2/a), focal distances, and equations of tangents and normals. A useful fact is that the sum of the focal distances of any point on the ellipse equals the length of the major axis, 2a. Always check whether the major axis lies along the x-axis or the y-axis before applying formulas. Practise the ellipse questions below, each with a step-by-step solution.

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The locus of the point of intersection of tangents to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ which meet at right angles is

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The locus of the point of intersection of tangents to the ellipse $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ which meet at right angles is