Name: Parametric Equation of an Ellipse
Uploaded: 2019-07-19
Duration: 7 min 12 s
Description: In this video, you will understand the parametric equation of an ellipse with examples, which will help you solve questions in the exam. Watch and learn

Question

Prove that the difference of the squares of the perpendiculars drawn from the centre upon parallel tangents to two given confocal ellipse is constant.

Accepted Answer

Let the given confocal ellipse be standard ellipses $E_{1}$ and $E_{2}$ , where

$E_{1} :$ $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b) . . . (i)$

Here, foci $(\pm a e, 0)$ or $(\pm \sqrt{a^{2} - b^{2}}, 0)$ and centre $(0, 0)$ .

$(∵ e = \sqrt{1 - \frac{b^{2}}{a^{2}}})$

and $E_{2} :$ $\frac{x^{2}}{A} + \frac{y^{2}}{B} = 1$

Here, foci $(\pm \sqrt{A - B}, 0)$

Two conics are said to be confocal when they have both foci common.

These foci are the same so $a^{2} - b^{2} = A - B$

$\Rightarrow A - a^{2} = B - b^{2} = k$ (Say)

$\Rightarrow A = a^{2} + k, B = b^{2} + k$

So, the equation of a confocal ellipse is

$\frac{x^{2}}{a^{2} + k} + \frac{y^{2}}{b^{2} + k} = 1 . . . (ii)$

For an ellipse

$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

Let $l x + m y + n = 0$ is tangent, then we have $n^{2} = a^{2} l^{2} + b^{2} m^{2} . . . (iii)$

If the equation of tangent is in normal form of line, i.e.,

$x \cos θ + y \sin θ = P$ , where $P$ is the perpendicular distance from the origin to the line (here origin is centre of the ellipse)

Using Equation $(iii)$ , we get

$P^{2} = a^{2} \cos^{2} θ + b^{2} \sin^{2} θ . . . (iv)$

Now for ${II}^{nd}$ conic, we have equation of tangent which is parallel to the tangent of the ellipse.

So, $x \cos θ + y \sin θ = P_{1}$

Here, $P_{1}$ , also the perpendicular distance from centre to tangent.

$P_{1}^{2} = (a^{2} + k) \cos^{2} θ + (b^{2} + k) \sin^{2} θ . . . (v)$

Now difference of $(iv)$ & $(v)$ , we get,

$P_{1}^{2} - P^{2} = (a^{2} + k) \cos^{2} θ + (b^{2} + k) \sin^{2} θ - a^{2} \cos^{2} θ - b^{2} \sin^{2} θ$

$= k \cos^{2} θ + k \sin^{2} θ$

$= k$ (constant)

S L Loney Solutions for Chapter: Miscellaneous Propositions, Exercise 1: EXAMPLES XLVII

S L Loney Mathematics Solutions for Exercise - S L Loney Solutions for Chapter: Miscellaneous Propositions, Exercise 1: EXAMPLES XLVII

Questions from S L Loney Solutions for Chapter: Miscellaneous Propositions, Exercise 1: EXAMPLES XLVII with Hints & Solutions

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