Question

Find the equation of the tangent and the normal to the following curve at the indicated point  y = x² at (0, 0) ?

Answer 1

$$y=x^2$$

$$\text{ Differentiating both sides w.r.t.x,}$$

$$\frac{dy}{dx}=2x$$

$$\text{ Given }(x_1,y_1)=(0,0)$$

$$\text{ Slope of tangent,}m={\left(\frac{dy}{dx}\right)}_{(0,0)}=2\left(0\right)=0$$

$$\text{ Equation of tangent is },$$

$$y-y_1=m(x-x_1)$$

$$\Rightarrow y-0=0(x-0)$$

$$\Rightarrow y=0$$

$$\text{ Equation of normal is,}$$

$$\Rightarrow y-y_1=\frac{-1}{m}(x-x_1)$$

$$\Rightarrow y-0=\frac{-1}{0}(x-0)$$

$$\Rightarrow x=0$$