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Normal of a Curve y = f(x)

A tangent is ...

Question

A tangent is drawn to the curve $y = x^{3} + 3 x^{2} + 1$ at $(h, k); h > 0$ . If the equation of tangent is $y = 9 x - 4$ then the value of $h + k$ is

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Solution

The correct option is B $6$
Equation of tangent
$y = 9 x - 4$
Slope of tangent $m = 9$
Given curve
$y = x^{3} + 3 x^{2} + 1$
Differentiating w.r.t. $x$ ,
$y^{'} = 3 x^{2} + 6 x \Rightarrow 9 = 3 h^{2} + 6 h \Rightarrow h^{2} + 2 h - 3 = 0 \Rightarrow (h + 3) (h - 1) = 0 \Rightarrow h = 1; - 3$
$∴ h = 1$
Putting in the tangent equation
$k = 9 - 4 \Rightarrow k = 5$
Hence $h + k = 1 + 5 = 6$

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