Video Transcript
Find the equation of the tangent to
nine 𝑦 squared equals negative seven 𝑥 plus nine that has gradient seven over
18.
So we’ve got to find the equation
of the tangent, and we’re given the slope as seven over 18. So, we’re going to want to use the
equation 𝑦 minus 𝑦 nought equals 𝑚 multiplied by 𝑥 minus 𝑥 nought, where this
line goes through 𝑥 nought, 𝑦 nought, with slope 𝑚. Firstly, we’ll find the point at
which the line is tangent to the curve. So, that will be 𝑥 nought, 𝑦
nought. And secondly, we find the point
slope equation of the tangent line. So, we need to find 𝑥 nought and
𝑦 nought. And the way we’ll do this is by
differentiating the equation.
So we’re going to need to do this
implicitly with respect to 𝑥. So, we want to differentiate both
sides with respect to 𝑥. Now, for the right-hand side, we
recall that the derivative of 𝑎𝑥 is just 𝑎, and constants differentiate to
zero. And so, negative seven 𝑥 plus nine
differentiates to negative seven. The derivative of the left-hand
side is a bit trickier. This is because we’re
differentiating this function of 𝑦, with respect to 𝑥. So, we have to apply the chain
rule.
The chain rule says that d𝑓 by d𝑥
is equal to d𝑓 by d𝑦 multiplied by d𝑦 by d𝑥. The function we want to
differentiate is nine 𝑦 squared. So, if we let 𝑓 equal nine 𝑦
squared, and so d𝑓 by d𝑥 equals d𝑓 by d𝑦, and this is nine 𝑦 squared
differentiated with respect to 𝑦. We can do this because we know the
power rule of differentiation. So, d𝑓 by d𝑦 is 18𝑦. And that’s multiplied by d𝑦 by
d𝑥. And now we can put this back into
our working out.
So, we have the 18𝑦 multiplied by
d𝑦 by d𝑥 equals negative seven. So now, we have an equation
involving d𝑦 by d𝑥. So, let’s rearrange to make d𝑦 by
d𝑥 the subject. We divide both sides by 18𝑦. And we get d𝑦 by d𝑥 equals
negative seven over 18𝑦.
Now remember, in the question we
were told that the slope d𝑦 by d𝑥 is equal to seven over 18. So, replacing d𝑦 by d𝑥 by seven
over 18, we have that seven over 18 equals negative seven over 18𝑦. And, solving for 𝑦 gives us that
𝑦 equals negative one. Substituting this back into the
original equation, nine 𝑦 squared equals negative seven 𝑥 plus nine, gives us nine
negative one squared equals negative seven 𝑥 plus nine. Remember that negative one squared
is just one. And then solving for 𝑥, we find
that 𝑥 equals zero.
Remember that these values that we
found are the coordinates for the point at which the line is tangent to the
curve. So, this is our 𝑥 nought, 𝑦
nought. And now, we need to use the point
slope equation to find the tangent line.
Applying the point slope equation
with 𝑥 nought equal to zero. 𝑦 nought equal to negative
one. And the slope 𝑚 equal to seven
over 18. We have that the tangent line is 𝑦
minus negative one equals seven over 18 multiplied by 𝑥 minus zero. 𝑦 minus negative one is just 𝑦
plus one. And, expanding the brackets on the
right-hand side gives us that 𝑦 plus one equals seven over 18𝑥. We can gather all of our terms onto
one side of the equation. And, we can multiply all our terms
by 18 to get rid of the fraction to give us 18𝑦 minus seven 𝑥 plus 18 equals
zero. And, that gives us our final
answer.
