For the curve $7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0$ which of the following is true:

An hyperbola with eccentricity

\sqrt{3}

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An hyperbola with directrix

2 x + y - 1 = 0

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An hyperbola with focus

(1, 2)

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All of these

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Solution

The correct option is B All of these
Given $7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0$
$\Rightarrow 7 x^{2} - 2 y^{2} = - [2 \times 2 \times 3 x y - 2 x + 14 y - 22] = - 3 {(2 x + y - 1)}^{2} + 12 x^{2} + 3 y^{2} + 25 - 10 x - 20 y \Rightarrow 7 x^{2} - 2 y^{2} - 12 x^{2} - 3 y^{2} - 25 = - 3 {(2 x + y - 1)}^{2} - 10 x - 20 y \Rightarrow - 5 [x^{2} + y^{2} - 2 x - 4 y + 5] = - 3 {(2 x + y - 1)}^{2}$
$\Rightarrow {(x - 1)}^{2} + {(y - 2)}^{2} = \frac{3}{5} {(2 x + y - 1)}^{2} \Rightarrow {(x - 1)}^{2} + {(y - 2)}^{2} = 3 {(\frac{2 x + y - 1}{\sqrt{2^{2} + 1}})}^{2}$
$\Rightarrow \sqrt{{(x - 1)}^{2} + {(y - 2)}^{2}} = \sqrt{3} (\frac{2 x + y - 1}{\sqrt{5}}) \Rightarrow \frac{P S}{P M} = \sqrt{3} > 1$
Therefore, the given equation represents a hyperbola with eccentricity $\sqrt{3}$
Directrix is $2 x + y - 1 = 0$
Focus is $(1, 2)$

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Similar questions

7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

(5, 5)

7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22 = 0

S = 7 x^{2} - 2 y^{2} + 12 x y - 2 x + 14 y - 22

(5, 5)

S_{1} < 0

Find the equation of the hyperbola whose directrix is $2 x + y = 1$ focus $(1, 2)$ and eccentricity $\sqrt{3}$

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