Q106 / 160 · AP EAMCET 2024 (19 May Shift 2) · Mathematics

If the equation of tangent at $(2,3)$ on $y^2=a x^3+b$ is $y=4 x-5$, then the value of $a^2+b^2=$

A
51
Correct answer
B
53
C
28
D
25

Solution

Given the curve, $y^2=a x^3+b$ $\Rightarrow 2 y \frac{d y}{d x}=3 a x^2 \Rightarrow \frac{d y}{d x}=\frac{3 a x^2}{2 y}$ at $(2,3)$, $\frac{d y}{d x}=\frac{12 a}{6}=2 a$ Since, slope of tangent to the line $y=4 x-5$ is 4 $\begin{aligned} & \text { So, } 2 a=4 \Rightarrow a=2 \\ & b=9-16=-7 \quad \therefore a^2+b^2=49+4=53 \end{aligned}$

Hence, the correct answer is option: 'B'.

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