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Angle between Two Lines

Find the equa...

Question

Find the equation of hyperbola whose focus is $(1, 1)$ , directrix is $3 x + 4 y + 8 = 0$ and eccentricity $= 2$

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Solution

Given that : $e = 2, S (1, 1)$ and equation of directrix is $3 x + 4 y + 8 = 0$

Let a point be $P (x, y)$ , such that

$S P = e P M$ , where $P M$ is perpendicular distance from $P (x, y)$ to directrix

$\Rightarrow \sqrt{(x - 1)^{2} + (y - 1)^{2}} = 2 \times ∣ ∣ ∣ ∣ \frac{3 x + 4 y + 8}{\sqrt{3^{2} + 4^{2}}} ∣ ∣ ∣ ∣$

$\Rightarrow \sqrt{(x - 1)^{2} + (y - 1)^{2}} = \frac{2}{5} | 3 x + 4 y + 8 |$

Squaring both sides, we get
$\Rightarrow (x - 1)^{2} + (y - 1)^{2} = \frac{4}{25} (3 x + 4 y + 8)^{2}$
$\Rightarrow 25 (x^{2} - 2 x + 1 + y^{2} - 2 y + 1) = 4 (9 x^{2} + 16 y^{2} + 64 + 24 x y + 64 y + 48 x)$
$\Rightarrow 25 x^{2} - 50 x + 25 y^{2} - 50 y + 50 = 36 x^{2} + 64 y^{2} + 256 + 96 x y + 256 y + 192 x$
$\Rightarrow 11 x^{2} + 39 y^{2} + 96 x y + 242 x + 306 y + 206 = 0$

$∴$ Equation of hyperbola is
$11 x^{2} + 96 x y + 39 y^{2} + 242 x + 306 y + 206 = 0$

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Similar questions

\sqrt{3}

\frac{4}{3}

Find the equaion of the hyperbola whose

(i) focus is (0,3), directrix is x+y-1=0 and eccentricity=2

(ii) foucs is (1,1), directrix is 3x+4y+8=0 and eccentricity=2

(iii) focus is (1,1), directrix is 2x+y=1 and 3ccentricity= $\sqrt{3}$

(iv) focus is 92,-1), firectrix is 2x+3y=1 and eccentricity =2.

(v) focus is (a,0).directrix is 2x-y+a=0 and eccentricity = $\frac{4}{3}$

(vi) focus is (2,2), directiex is x+y=9 and eccentricity =2.

(1, 1)

2 x + y = 1

= \sqrt{3}

(a, 0)

2 x - y + a = 0

= \frac{4}{3}

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