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The equation of the tangent

Question:

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

Solution:

finding the slope of the tangent by differentiating the curve

$2 y \frac{d y}{d x}=3 a x^{2}$

$\frac{d y}{d x}=\frac{3 a x^{2}}{2 y}$

$\mathrm{m}$ (tangent) at $(2,3)=2 \mathrm{a}$

equation of tangent is given by $y-y_{1}=m($ tangent $)\left(x-x_{1}\right)$

now comparing the slope of a tangent with the given equation

$2 a=4$

$a=2$

now $(2,3)$ lies on the curve, these points must satisfy

$3^{2}=2 \times 2^{3}+b$

$b=-7$