# How do you find the equations for the normal line to x^2+y^2=20 through (2,4)?

Apr 23, 2017

$y = 2 x$

#### Explanation:

$y - 0 = \frac{4 - 0}{2 - 0} \left(x - 0\right)$

$y = 2 x$

:)))

Apr 23, 2017

The equation is $y = 2 x$.

#### Explanation:

The derivative of this relation is given by

$2 x + 2 y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = 0$

$2 y \left(\frac{\mathrm{dy}}{\mathrm{dx}}\right) = - 2 x$

$\frac{\mathrm{dy}}{\mathrm{dx}} = - \frac{x}{y}$

The slope of the tangent line is

${m}_{\text{tangent}} = - \frac{2}{4} = - \frac{1}{2}$

Therefore, the slope of the normal line is $2$.

$y - 4 = 2 \left(x - 2\right)$

$y - 4 = 2 x - 4$

$y = 2 x$, as obtained in the other answer

Hopefully this helps!