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

1 Answer
Dec 13, 2016

# y=-1/4x+9/2#

Explanation:

The gradient of the tangent to a curve at any particular point is give by the derivative of the curve at that point. The normal is perpendicular to the tangent, so the product of their gradients is #-1#

so If #y=x^2# then differentiating wrt #x# gives us:

#dy/dx = 2x#

When #x=2 => y=2^2=4# (so #(2,4)# lies on the curve)
and #dy/dx=(2)2=4#

So the normal we seek passes through #(2,4)# ad has gradient #-1/4# so using #y-y_1=m(x-x_1)# the equation we seek is;

# y-4=-1/4(x-2) #
# :. y-4=-1/4x+1/2#
# :. y=-1/4x+9/2#

We can confirm this graphically:
enter image source here