Name: Classification of Conic Based on Eccentricity
Uploaded: 2019-07-19
Duration: 9 min 45 s
Description: We know that eccentricity is a measure of how much a conic section deviates from being circular. In this video, we will learn about the eccentricity of diffe...

Question

Find the equation of a conic whose focus is at

(1, 0)

and directrix is

12 x + 5 y + 1 = 0

. Also,

e = 1 / \sqrt{2}

.

Accepted Answer

Given focus is $(1, 0)$ , the equation of the corresponding directrix is $12 x + 5 y + 1 = 0$ and eccentricity $(e) = \frac{1}{\sqrt{2}}$

If $S$ is the focus, $P$ is any point on the conic section and $P M$ is the length of the perpendicular from $P$ on the directrix, then by definition, we know that

$\frac{S P}{P M} = e$

Let the coordinates of any point $P$ be $(h, k)$ then by the distance formula, we have

$\sqrt{{(h - 1)}^{2} + k^{2}} = \frac{1}{\sqrt{2}} |\frac{12 h + 5 k + 1}{\sqrt{{(12)}^{2} + 5^{2}}}|$

$\Rightarrow \sqrt{h^{2} - 2 h + 1 + k^{2}} = \frac{1}{\sqrt{2}} |\frac{12 h + 5 k + 1}{\sqrt{169}}|$

$h^{2} - 2 h + 1 + k^{2} = \frac{144 h^{2} + 25 k^{2} + 1 + 120 h k + 10 k + 24 h}{338}$

$\Rightarrow 338 h^{2} - 676 h + 338 + 338 k^{2} = 144 h^{2} + 25 k^{2} + 120 h k + 10 k + 24 h + 1$

$\Rightarrow 194 h^{2} - 700 h - 120 h k + 313 k^{2} - 10 k + 337 = 0$

Now, replacing $(h, k)$ by $(x, y)$ , we get the required equation of the conic.

$\Rightarrow 194 x^{2} - 700 x - 120 x y + 313 y^{2} - 10 y + 337 = 0$ .

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