Video Transcript
Find the equation of the tangent to
the curve 𝑦 equals 𝑥 cubed plus nine 𝑥 squared plus 26𝑥 that makes an angle of
135 degrees with the positive 𝑥-axis.
So we’ve been asked to find the
equation of a tangent to a particular curve, which we know we can do using
differentiation and the general equation of a straight line. But what does it mean when it says
this tangent makes an angle of 135 degrees with the positive 𝑥-axis? Let’s consider a sketch. Well, it will look something like
this. The tangent here is shown in
pink. And we can see that when intersects
with the 𝑥-axis, the angle between the positive 𝑥-axis and the tangent is 135
degrees.
In order to apply the general
equation of a straight line 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one, we either
need to know the slope 𝑚 of our line or the coordinates of a point 𝑥 one, 𝑦 one
which lies on the line. So how does knowing that our
tangent makes an angle of 135 degrees with the positive 𝑥-axis help with
determining either of those? Well, the angle on the other side
of this line will be 45 degrees because we know that angles on a straight line sum
to 180 degrees. We can sketch in a right-angled
triangle below this line and recall that the slope of a line is change in 𝑦 over
change in 𝑥. That’s the vertical height of this
triangle divided by the horizontal distance. But in that right triangle, those
sides are the opposite and adjacent in relation to the angle of 45 degrees. So we’re dividing the length of the
opposite by the length of the adjacent.
As the line is sloping downwards
though, that vertical change is actually the negative of the value of the
opposite. So we have that the slope is equal
to negative opposite over adjacent. Opposite divided by adjacent
defines the tangent ratio. So in fact, this is equal to
negative tan of 45 degrees. And tan of 45 degrees is just
one. So by considering this right-angled
triangle, we found that the slope of this line is negative one. So, we found the slope of our
tangent. But we don’t yet know the
coordinates of the point on the curve where this tangent is being drawn. To find this, we need to find the
point on the curve where the gradient or slope is equal to negative one.
We begin by differentiating the
equation of the curve with respect to 𝑥 and applying the power rule of
differentiation, giving d𝑦 by d𝑥 equals three 𝑥 squared plus 18𝑥 plus 26. We then set this expression equal
to negative one to find the 𝑥-coordinate of the point on the curve, where the
gradient is negative one. This simplifies to three 𝑥 squared
plus 18𝑥 plus 27 equals zero. And then dividing through by three
gives 𝑥 squared plus six 𝑥 plus nine equals zero. We should notice that this is, in
fact, a perfect square. We can write it as 𝑥 plus three
all squared. Solving this equation then, this
means that 𝑥 plus three must be equal to zero. And so, 𝑥 is equal to negative
three.
Next, we need to find the value of
𝑦 when 𝑥 is equal to negative three, which we do by substituting negative three
into the equation of the curve. And it gives negative 24. We now know that this tangent has a
slope of negative one at the point negative three, negative 24. All that’s left is to substitute
into the general equation of the straight line. 𝑦 minus negative 24 equals
negative one multiplied by 𝑥 minus negative three. That all simplifies to 𝑦 plus 𝑥
plus 27 equals zero.
The key steps in this question then
were to use some trigonometric reasoning to identify that if a line makes an angle
of 135 degrees with the positive 𝑥-axis. Then, its gradient or slope is
equal to negative tan 45 degrees, which is equal to negative one. We then use the gradient function
of the curve to identify the 𝑥-value at which the gradient was equal to negative
one. We found the corresponding 𝑦-value
by substituting into the equation of the curve and then finally used the general
equation of a straight line 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one to find the
equation of this tangent.
