Question

An equation of the tangent to curve $y=x^4$ from the point $(2,0)$on the curve is

• A. $y=0$
• B. $x=0$
• C. $x+y=0$
• D. None of these

Answer 1

The correct option is A

$y=0$

Explanation for the correct option:

Find the equation of the tangent plane :

Let the point of contact be $(h,k)$ where $k=h^4$

The tangent is $y–k=m(x–h)$ where $m=4h^3$,which passes through $(2,0).$

$$\begin{gathered} \therefore –k=4h^3(2–h) \\ \Rightarrow -h^4=8h^3-4h^4 \\ \Rightarrow 3h^4=8h^3 \\ \Rightarrow h^3(3h-8)=0 \\ \Rightarrow h=0or\frac{8}{3} \\ \Rightarrow k=0or{\left(\frac{8}{3}\right)}^4 \end{gathered}$$

Therefore the point of contact are $(0,0)$ and $\left(\frac{8}{3},{\left(\frac{8}{3}\right)}^4\right).$

The equation of tangents are $y=0$ and $y-{\left(\frac{8}{3}\right)}^4=4{\left(\frac{8}{3}\right)}^3\left(x-\left(\frac{8}{3}\right)\right)$.

Hence, the correct option is (A).