### Video Transcript

Determine the points on a curve π₯
squared plus π¦ squared plus π₯ plus ~~π₯~~ [eight] π¦ that equals zero at
which the tangent is perpendicular to the line seven π¦ plus four π₯ plus π equals
zero.

So the first part of this question
weβre actually gonna have a look at is bit where it says the tangent is
perpendicular to the line and that line is seven π¦ plus four π₯ plus π equals
zero. Okay, so letβs use this first.

So if we have our line, then what
we wanna do is we actually want to rearrange it. And we want to rearrange it into
the form π¦ equals ππ₯ plus π because that way as you can see here π is going to
be the slope and the plus π will be the π¦-intercept. And in this case itβs actually the
slope that weβre looking for. So if we subtract four π₯ and π,
we get seven π¦ is equal to negative four π₯ minus π. And then if we divide both sides by
seven, we get π¦ is equal to negative four π₯ minus π over seven.

So Iβm just gonna rewrite this to
make it more clear what the slopeβs going to be. Then, we get π¦ is equal to
negative four over seven π₯ minus π over seven. So therefore, we can say that the
slope is gonna be our π. So itβll be negative four over
seven.

Okay, so we found the slope. But why is this useful? Well, itβs actually this word here
that makes this useful βperpendicularβ because actually what weβre looking for is
weβre looking for the points on the curve where the tangent is perpendicular to our
line. So if itβs perpendicular, then
therefore the slope is gonna be the negative reciprocal. So therefore, we can say the slope
at the points is going to be seven over four. And this is because this is a
negative reciprocal of negative four over seven.

Okay, fantastic, now we know what
slope weβre looking for. We need to now actually work out
the slope function of our curve. So if we have the function π₯
squared plus π¦ squared plus π₯ plus eight π¦ is equal to zero, then what weβre
gonna do is actually gonna differentiate this to find our slope function. In order to do this, weβre gonna
use implicit differentiation.

So the first stage of implicit
differentiation, which I wouldnβt normally write up, but I just put here to show
actually whatβs happening, is we actually differentiate each of our terms with
respect to π₯ which is gonna give us two π₯ plus two π¦ ππ¦ ππ₯ plus one plus
eight ππ¦ ππ₯ equals zero. Just to remind us how we got the
second and fourth terms, if we have a term thatβs in π¦ β so itβs a functional term
in π¦ β and we want to differentiate it with respect to π₯, then this is equal to
the same functional term differentiated with respect to π¦ multiplied by ππ¦
ππ₯. And this comes from actually
adapting the chain rule.

So therefore, if we look at our
second term, if we had differentiating π¦ squared with respect to π₯, then what we
can say is this going to be equal to two π¦ because if you differentiate π¦ squared
with respect to π¦, you get two π¦ because you multiply the coefficient by the
exponents β so two by one which gives us two β and then you reduce the exponent by
one, so just leaving us with π¦. And then, we multiply this by ππ¦
ππ₯.

Okay, fab, we now know where these
have come from, we can move on to the next stage. Well, the next stage is actually to
make ππ¦ ππ₯ the subject because as I said before this is our slope function and
this is what weβre looking to find. So therefore, weβre gonna get two
π¦ ππ¦ ππ₯ plus eight ππ¦ ππ₯ equals negative two π₯ minus one. And now actually as we have ππ¦
ππ₯ as a factor in both of our terms on the left-hand side, we can actually
factor. So therefore, we get ππ¦ ππ₯
multiplied by two π¦ plus eight equals negative two π₯ minus one.

So therefore, to finally find our
slope function, all we need to do is divide both sides by two π¦ plus eight. So therefore, we can say ππ¦ ππ₯
β so our slope function β is equal to negative two π₯ minus one over two π¦ plus
eight.

Okay, fantastic, so weβve now got
this. Whatβs the next stage? Well, from early, we actually
calculated what the slope would be on the points on our curve where the tangent is
perpendicular to the line seven π¦ plus four π₯ plus π equals zero. So what we can do is we can
actually substitute in our value for the slope into our slope function and then
solve to find the equation of our tangent.

So now, weβve actually substituted
in seven over four for our slope because we know thatβs the slope that we want at
those points. So we can say that negative two π₯
minus one over two π¦ plus eight equals seven over four. So therefore, we can say that four
multiplied by negative two π₯ minus one is equal to seven multiplied by two π¦ plus
eight which is gonna give us negative eight π₯ minus four equals 14π¦ plus 56, which
if we rearrange gives us 14π¦ plus eight π₯ plus 56 plus four equals zero. And then if we collect together all
the terms, we have 14π¦ plus eight π₯ plus 60 equals zero.

Then, one more stage, we just
divide by two to make it a bit easier to deal with. So now, we have the equation of the
tangent at the points that weβre looking for and that is seven π¦ plus four π₯ plus
30 is equal to zero.

Now, this is the stage where we
actually look back at the question because weβve done lots of things and found lots
of different parts here. But what is it the question
actually wants? Well, the question wants the points
on the curve. So in that case, we need to find
out π₯- and π¦-values. And to do that, weβre gonna set up
a simultaneous equation.

So as you can see the equations
that Iβm going to solve simultaneously are actually the function of our curve, so π₯
squared plus π¦ squared plus π₯ plus eight π¦ equals zero, and the line which we
found which is the tangent which is seven π¦ plus four π₯ plus 30 equals zero. And weβve done this because at the
points where they actually meet each other are the points that weβre looking
for.

Well, the first stage is actually
to rearrange equation two to make π₯ the subject. So weβre gonna get four π₯ equals
negative 30 minus seven π¦, so subtract the 30 and seven π¦ from each side of the
equation. So then, if we divide both sides of
our equation by four, we get π₯ is equal to negative 30 minus seven π¦ over
four. So great weβve now got this and we
can actually substitute it back into equation one to help us find what π¦ is. So when we do this, we get negative
30 minus seven π¦ over four all squared because weβve substituted that value of
negative 30 minus seven π¦ over four in for π₯ plus π¦ squared plus negative 30
minus seven π¦ over four plus eight π¦ equals zero.

And then, we expand our
parentheses. So we get 900 plus 420π¦ plus 49π¦
squared over 16 plus π¦ squared plus negative 30 minus seven π¦ over four plus eight
π¦ equals zero. So then, what weβre gonna do is
multiply through by 16 just to remove the fractions which gives us 900 plus 420π¦
plus 49π¦ squared plus 16π¦ squared minus 120 minus 28π¦ plus 128π¦ equals zero. So now, weβre gonna simplify and
collect like terms which gives us 65π¦ squared plus 520π¦ plus 780 equals zero.

Okay, so we could actually look to
factor at this point. But again, can we make it
simpler? Always try to simplify and yes, we
can. We can actually divide three by 65
which leaves us with a much more manageable π¦ squared plus eight π¦ plus 12 equals
zero. So great, we can now factor this to
actually find out our values of π¦. So we get π¦ plus six multiplied by
π¦ plus two equals zero. So therefore, π¦ is equal to
negative six or negative two. So fantastic, weβve actually now
found out our π¦-coordinates of the points on the curve that weβre looking for. Now, letβs substitute these values
back into one of our equations to find out the value of π₯.

So now, what weβre gonna do is
actually substitute our π¦-values into the equation we made for π₯ to find our
π₯-values. So if we start with π¦ equals
negative six, weβre gonna get π₯ is equal to negative 30 minus seven multiplied by
negative six over four which gives us a value of π₯ of three. And then, if we substitute in π¦
equals negative two, weβre gonna get π₯ is equal to negative 30 minus and then seven
multiplied by negative two all over four which is gonna give us an π₯-value of
negative four.

So therefore, we can say that the
points on the curve π₯ squared plus π¦ squared plus π₯ plus eight π¦ equals zero at
which the tangent is perpendicular to the line seven π¦ plus four π₯ plus π equals
zero are three, negative six and negative four, negative two.