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An equation of the tangent to the curve $y=x^4$ from the point $(2,0)$ not on the curve is
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Verified Answer
The correct answer is:
$y=0$
Let the point of contact be $(h, k)$, where $k=h^4$.
Tangent is $y-k=4 h^3(x-h),\left[0 \frac{d y}{d x}=4 x^3\right]$
It passes through $(2,0), \therefore-k=4 h^3(2-h)$ $\Rightarrow h=0$ or $8 / 3, \therefore k=0$ or $(8 / 3)$
$\therefore$ Points of contact are $(0,0)$ and $\left(\frac{8}{3},\left(\frac{8}{3}\right)^4\right)$
$\therefore$ Equation of tangents are
$y=0 \text { and } y-\left(\frac{8}{3}\right)^4=4\left(\frac{8}{3}\right)^3\left(x-\frac{8}{3}\right)$
Tangent is $y-k=4 h^3(x-h),\left[0 \frac{d y}{d x}=4 x^3\right]$
It passes through $(2,0), \therefore-k=4 h^3(2-h)$ $\Rightarrow h=0$ or $8 / 3, \therefore k=0$ or $(8 / 3)$
$\therefore$ Points of contact are $(0,0)$ and $\left(\frac{8}{3},\left(\frac{8}{3}\right)^4\right)$
$\therefore$ Equation of tangents are
$y=0 \text { and } y-\left(\frac{8}{3}\right)^4=4\left(\frac{8}{3}\right)^3\left(x-\frac{8}{3}\right)$
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