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find the equation of the hyperbola which...

find the equation of the hyperbola which passes through the points of intersection of the line x+y=5 and hyperbola `2x^2-3y^2-6=0`and is also passes through (3,4).

A

`2x-y-2=0`

B

`3x-2y=0`

C

`2x-3y+10=0`

D

`x-2y+8=0`

Text Solution

Verified by Experts

For the ellipse `(x^(2))/(25)+(y^(2))/(9)=1`, we have `a=5`, `b=3`.
`:.e=sqrt(1-(9)/(25))=(4)/(5)`
So, the coordinates of the foci are `(+-4,0)`. These are also foci of the hyperbola `(x^(2))/(a^(2))-(y^(2))/(b^(2))=1` whose eccentricity is `2`.
`:.ae=4impliesa=2`
Now, `b^(2)=a^(2)(e^(2)-1)impliesb^(2)=4(4-1)=12`
So, the equation of the hyperbola is `(x^(2))/(4)-(y^(2))/(12)=1`.
Clearly , point `(4,6)` lies on it. The equation of tangent to this hyperbola at `(4,6)` is
`(4x)/(4)-(6y)/(12)=1` or, `2x-y=2`
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