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If the foci of a hyperbola lie on y=x an...

If the foci of a hyperbola lie on `y=x` and one of the asymptotes is `y=2x ,` then the equation of the hyperbola, given that it passes through (3, 4), is (a)`x^2-y^2-5/2x y+5=0` (b) `2x^2-2y^2+5x y+5=0` (c) `2x^2+2y^2-5x y+10=0` (d) none of these

Text Solution

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The correct Answer is:
`2x^(2)+2y^(2)-5xy+10=0`
The foci of the hyperbola lie on y = x. So, the major axis is y = x.
The major axis of hyperbola bisects the asymptote.
Therefore, the equation of other asymptote is x = 2y.
Thus, the equation of hyperbola is `(y-2x)(x-2y)+k=0.`
Given that it passes through (3,4), we get `k=-10.`
Hence, the required equation is
`2x^(2)+2y^(2)-5xy+10=0`
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