### Video Transcript

Find an equation of the tangent to the curve ๐ฅ equals root ๐ก, ๐ฆ equals ๐ก squared minus two ๐ก at the point corresponding to the value ๐ก equals four.

So the first thing we want to do is actually find out what point on the curve is the tangent. And to do that, what weโre gonna do is actually substitute in ๐ก equals four to each of our parametric equations.

So weโre gonna start with our ๐ฅ-coordinate. And to find our ๐ฅ-coordinate, what weโre gonna do is substitute ๐ก equals four into ๐ฅ equals root ๐ก. And when we do that, we get that ๐ฅ is equal to the root four. So therefore, our ๐ฅ-value is gonna be equal to two.

Okay, great, so weโve got our first coordinate. So now, letโs find the ๐ฆ-coordinate. Well, the ๐ฆ-coordinate is gonna be equal to four squared minus two multiplied by four, which is gonna give us 16 minus eight. So therefore, weโre gonna get a final answer of ๐ฆ is equal to eight.

So therefore, we can say that the point on the curve that the tangent acts is actually two, eight. Okay, great, so now what? So what we need to do actually is actually find the equation of the tangent to the curve. So therefore, what we want to do first is find the slope at the point that weโve got on the curve. And the reason we want to do that is because at that point the slope of the curve and the slope of the tangent will be equal to each other.

So what we actually have is a pair of parametric equations. And we actually have a special relationship that tells us what the slope function is for parametric equations. So our slope function d๐ฆ d๐ฅ is equal to d๐ฆ d๐ก divided by d๐ฅ d๐ก. The reason itโs really useful is because it allows us to deal with each of parametric equations separately and then bring them together to find our slope.

So Iโm gonna to start with ๐ฅ equals root ๐ก. So what we want to do is actually differentiate this to find d๐ฅ d๐ก. But in order to make this easier, what we want to do first is actually rewrite this in exponent form. So we got ๐ฅ is equal to ๐ก to the power of a half. So therefore, d๐ฅ d๐ก is gonna be equal to a half ๐ก to the power of negative a half.

And to just remind us how we got that is cause when you differentiate, what you do is to multiply the coefficient which is one by the exponent which is a half. So we get a half and then itโs ๐ก to the power of and then you subtract one from the exponent. So a half minus one gives us negative a half.

Okay, great, we found d๐ฅ d๐ก. Now, letโs move on and find d๐ฆ d๐ก. So what we have is that ๐ฆ is equal to ๐ก squared minus two ๐ก. So weโre gonna differentiate this. And when we differentiate this, we get two ๐ก minus two. So great, what weโve done is actually found d๐ฅ d๐ก and d๐ฆ d๐ก. So we can actually put it back into our form to find d๐ฆ d๐ฅ.

So therefore, what we do is when we substitute this back in to our formula for d๐ฆ d๐ฅ, what weโre gonna get is that our slope function d๐ฆ d๐ฅ is equal to two ๐ก minus two because thatโs our d๐ฆ d๐ก divided by a half ๐ก to the power of negative a half cause thatโs our d๐ฅ d๐ก. So therefore, this is gonna be equal to two ๐ก minus two multiplied by two ๐ก to the power of a half. And weโve got that because actually if you divide by a fraction, thatโs the same as multiplying by the reciprocal of that fraction.

So now, what we do is we actually substitute in ๐ก equals four to find out the value of our slope. So therefore, we can say that d๐ฆ d๐ฅ is equal to two multiplied by four minus two multiplied by two root four, which is gonna be equal to six multiplied by four which gives us 24.

Okay, great, so weโve now found the slope at the point two, eight. But why has this been useful? Well, itโs useful because actually weโre trying to find the equation of the tangent to the curve. So thatโs gonna be a straight line. And the general form for a straight line is ๐ฆ equals ๐๐ฅ plus ๐, where ๐ is our slope and ๐ is our ๐ฆ-intercept.

So therefore, we can say that ๐ฆ is gonna be equal to 24๐ฅ plus ๐. And thatโs because our slope was 24. But now, what we need to do is actually find out what ๐ is. And to find out ๐, we can actually substitute in our values from our points, so our ๐ฅ- and ๐ฆ-coordinates.

So therefore, when we do this, we get eight โ because thatโs what ๐ฆ is equal to โ is equal to 24 multiplied by two because ๐ฅ is equal to two then plus ๐. So therefore, eight is equal to 48 plus ๐. So now, if we actually actually subtract 48 from each side of the equation, we get ๐ is equal to negative 40.

So therefore, if we actually substitute this into our ๐ฆ equals ๐๐ฅ plus ๐ along with our 24 for our ๐, weโre gonna get the equation of the tangent to the curve ๐ฅ equals root ๐ก, ๐ฆ equals ๐ก squared minus two ๐ก at the point corresponding to the value ๐ก equals four, which is the point two, eight, is going to be ๐ฆ equals 24๐ฅ minus 40.