The equation of the hyperbola whose asymptotes are 2 x - y -3 and 3 x - y -7 and passing though the point 1-1 isA- 6 x2-x y-y2-20 x-4 y-12-0- 6 x 2- xy - y 2-23 x -4 y -15-0- 6 x 2- xy - y 2-20 x -4 y -20-0D- 6 x 2- xy - y 2-23 x -4 y -19-0

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Standard XII

Mathematics

Asymptotes

The equation ...

Question

The equation of the hyperbola whose asymptotes are $2 x - y = 3$ and $3 x + y = 7$ and passing though the point $(1, 1)$ is

6 x^{2} - x y - y^{2} - 20 x + 4 y + 12 = 0

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6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0

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6 x^{2} - x y - y^{2} + 20 x - 4 y - 20 = 0

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6 x^{2} + x y + y^{2} - 23 x - 4 y + 19 = 0

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Solution

The correct option is B $6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0$
The combined equation of the asymptotes is $(2 x - y - 3) (3 x + y - 7) = 0$
$\Rightarrow 6 x^{2} - x y - y^{2} - 23 x + 4 y + 21 = 0$

Let the equation of the hyperbola be $6 x^{2} - x y - y^{2} - 23 x + 4 y + λ = 0,$

where $λ$ is a constant such that it represents hyperbola which passes through $(1, 1)$ ,
so $λ = 15.$

Hence, the equation of the hyperbola becomes $6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0.$

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Q. The equation of the hyperbola whose asymptotes are

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$\begin{matrix} Equation & Name of the curve 1) x^{2} - 2 x - y - 3 = 0 & P) Circle 2) x^{2} + 3 x y + 2 y^{2} - x - 4 y - 6 = 0 & Q) Parabola 3) x^{2} + y^{2} - 20 = 0 & R) Ellipse 4) 7 x^{2} + 7 y^{2} + 2 x y + 10 x - 10 y + 7 = 0 & S) Hyperbola 5) 6 x^{2} - x y - y^{2} - 23 x + 4 y + 15 = 0 & T) Pair of straight lines \end{matrix}$

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Q. The equation of the circle having centre (1, –2) and passing through the point of intersection of the lines 3x + y = 14 and 3x + 5y = 18 is
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Find the equation of director circle of the circle whose diameters are 2x - 3y + 12 = 0 and x + 4y - 5 = 0 and area is 154 square units.

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Asymptotes

Standard XII Mathematics

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The equation of the hyperbola whose asymptotes are 2x−y=3 and 3x+y=7 and passing though the point (1,1) is

The equation of the hyperbola whose asymptotes are $2 x - y = 3$ and $3 x + y = 7$ and passing though the point $(1, 1)$ is