### Video Transcript

List the equations of the normals
to π¦ equals π₯ squared plus two π₯ at the points where the curve meets the line π¦
minus four π₯ equals zero.

What is meant by the term normal in
this context? Well, we recall first of all that
the tangent to a curve has the same gradient as the curve at that point. The normal, however, passes through
that same point, but it is perpendicular to the tangent at that point. We can use properties of
perpendicular lines to deduce the relationship that exists between the gradient of
the tangent and the gradient of the normal to a curve at a given point. The product of the two gradients
will be equal to negative one and they will be negative reciprocals of one
another.

We must make sure that weβre clear
whether weβve been asked to find the equation of a tangent or a normal when weβre
answering questions like this. So now that we know what normals
are, letβs see how we can answer this question. Weβve been asked to list the
equations of the normals to a given curve at the point where this curve meets
another line. So our first step is going to be to
find these points of intersection.

We can rearrange the equation of
the line to give π¦ equals four π₯ and then set the two expressions for π¦ equal to
one another to give an equation in π₯ only. We can subtract four π₯ from each
side and then factor the resulting quadratic to give π₯ multiplied by π₯ minus two
is equal to zero. The two roots of this equation are
π₯ equals zero or π₯ equals two. So we know the π₯-coordinates of
our points of intersection. To find the corresponding
π¦-coordinates, we substitute each π₯-value back into the equation of the curve to
give π¦ equals zero when π₯ equals zero and π¦ equals eight when π₯ equals two.

So we now know the two points of
intersection. And we, therefore, know the
coordinates of one point that lies on each normal. But we need to determine the
gradient or slope of each normal. First, we can find the slope of
each tangent by differentiating π¦ with respect to π₯, giving dπ¦ by dπ₯ equals two
π₯ plus two. When π₯ equals zero, the slope will
be two. And when π₯ equals two, the slope
will be six. But remember, this is the slope of
the tangent, not the slope of the normal. To find the slope of each normal,
we need to take the negative reciprocal of the slope of each tangent. So the slope of our first normal is
negative a half and the slope of our second is negative one-sixth.

Finally, we can apply the formula
for the general equation of a straight line. For the first normal with a slope
of negative a half passing through the point zero, zero, we get the equation two π¦
plus π₯ equals zero. And for the second with a slope of
negative one-sixth passing through the point two, eight, we get the equation six π¦
plus π₯ minus 50 equals zero. So we found the equations of the
two normals. We must be really careful on
questions like this. Remember, the slope of the normal
is not the same as the slope of the tangent. Itβs equal to the negative
reciprocal of the slope of the tangent because the two lines are perpendicular to
one another.