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If the tangent to the curve 2y^(3)=ax^(2...

If the tangent to the curve `2y^(3)=ax^(2)+x^(3)` at the point (a,a) cuts off intercept ` alpha and beta ` on the co-ordinate axes , (where ` alpha^(2)+beta^(2)=61`) then ` a^(2)` equals ______

A

400

B

600

C

900

D

800

Text Solution

Verified by Experts

The correct Answer is:
C
We have, `2y^(3) = ax^(2) +x^(3)`
`rArr (dy)/(dx) = (2ax +3x^(2))/(6y^(2)) rArr ((dy)/(dx))_("(a, a)")=(5)/(6)`
The equation of tangent to the given curve at (a, a) is `y-a=(5)/(6) (x-a)`
`rArr` x intercept of the tangent `=alpha= (1)/(5)a`
and y intercept of the tangent `=beta =(1)/(6)a`
`rArr alpha^(2)+beta^(2) =61 =(a^(2))/(25) +(a^(2))/(36) =(61a^(2))/(900) rArr a^(2) =900`
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