Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 11

Find the Equation of the Ellipse in the Case: Focus is (0, 1), Directrix is X + Y = 0 and E = 1 2 . - Mathematics

Short Note

Find the equation of the ellipse in the case:

focus is (0, 1), directrix is x + y = 0 and e = $\frac{1}{2}$ .

Solution

$\text{ Let S(0, 1) be the focus and ZZ' be the directrix . }$
$\text{ Let P(x, y) be any point on the ellipse and let PM be the perpendicular from P on the directrix } .$
$\text{ Then by the definition, we have: }$
$SP = e \times PM$
$\Rightarrow SP = \frac{1}{2} \times PM$
$\Rightarrow 2SP = PM$
$\Rightarrow 4 \left( SP \right)^2 = {PM}^2$
$\Rightarrow 4\left[ \left( x \right)^2 + \left( y - 1 \right)^2 \right] = \left| \frac{x + y}{\sqrt{1^2 + \left( 1 \right)^2}} \right|^2$
$\Rightarrow 4\left[ x^2 + y^2 + 1 - 2y \right] = \frac{x^2 + y^2 + 2xy}{2}$
$\Rightarrow 8 x^2 + 8 y^2 + 8 - 16y = x^2 + y^2 + 2xy$
$\Rightarrow 7 x^2 + 7 y^2 - 2xy - 16y + 8 = 0$
$\text{ This is the required equation of the ellipse.}$

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APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 26 Ellipse
Exercise 26.1 | Q 2.1 | Page 22