The equation of the tangent at `(2,\ 3)` on the curve `y^2=a x^3+b` is `y=4x-5` . Find the values of `a` and `b` .

Explanation

The slope of the given line `y=4 x-5` is 4

`y^{2}=a x^{3}+b ldots(1)` `2 y frac{d y}{d x}=3 a x^{2}`

`Rightarrow frac{d y}{d x}=frac{3 a x^{2}}{2 y}`

Slope of tangent `=(frac{d y}{d x})_{(2,3)}=frac{12 a}{6}=2 a`

Given that

Slope of tangent `=` slope of given line


`2 a=4 `

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