Question

Find the equations of the tangent and the normal to the curve $x^2+y^2=5,$ where the tangent is parallel to the line $2x-y+1=0$ .

Answer 1

Slope of line $(-A/B)$ $2x-y+1=0$ is $\left(\frac{-2}{-1}\right)=2$

For curve, $x^2+y^2=5$, $2x+2y{y}^{\prime }=0$

$\Rightarrow y^{\prime }=\frac{-x}{y}$

as $y^{\prime }=$ slope, $\Rightarrow \frac{-x}{y}=2$

$\Rightarrow 2y=-x$

Putting in $x^2+y^2=5$,

$5y^2=5$

$\Rightarrow y=\pm 1$

$\Rightarrow x=\mp 2$

Slope of tangent, $y^{\prime }=\frac{-x}{y}=\frac{\pm 2}{\pm 1}=2$

Slope of normal, $\frac{-1}{y}=\frac{-1}{2}$

Equation of tangent $(y\mp 1)=2(x\pm 2)$

$\Rightarrow y-1=2(x+2)$ or $y+1=2(x-2)$

Equation of normal: $(y\mp 1)=\frac{-1}{2}(x\pm 2)$

$\Rightarrow y-1=-\frac{1}{2}(x+2)$

or $y+1=-\frac{1}{2}(x-2)$.