If the foci of a hyperbola lie on the line y=x, one asymptote is y=2x and it is passing through the point (3, 4), then ___________
Answers
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Given:
The foci of a hyperbola lie on the line y=x, one asymptote is y=2x and it is passing through the point (3, 4)
To find:
The required equation.
Solution:
From given, we have,
The foci of a hyperbola lie on the line y = x,
⇒ The equation of the transverse axis, y - x = 0
The one asymptote is y = 2x
⇒ The other asymptote is x = 2y
Therefore, the equation of the hyperbola is given as follows.
(y - 2x) (x - 2y) + k = 0
As it is passing through the point (3, 4), then substitute these values in the above equation to find the value of k.
So, we get,
(4 - 2 (3)) (3 - 2 (4)) + k = 0
(4 - 6) (3 - 8) + k = 0
(-2) (-5) + k = 0
10 + k = 0
k = -10
Substitute the value of k back in the equation to find the required equation.
So, we get,
(y - 2x) (x - 2y) + (-10) = 0
xy - 2y² - 2x² + 4xy - 10 = 0
5xy - 2y² - 2x² - 10 = 0
2x² + 2y² - 5xy + 10 = 0
Therefore, the required equation is 2x² + 2y² - 5xy + 10 = 0