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Hi, I'm Miss Davies.

In this lesson, we're going to be finding the equation of a tangent to a circle at a given point.

I've drawn onto this circle a radius and a tangent.

Here's a different radius and its tangent and here's another radius and tangent.

What's the same and what's different about these three sets of tangents and radii? All three pairs of a tangent and a radius meet at a perpendicular point.

The gradient of each of the tangents is different, and the gradient of each of the radii is different.

In this example, we've been asked to find the equation of the tangent to the circle at the point three, four.

Since a tangent and a radius that meet at a common point are perpendicular, their gradients must product to negative one.

We can see that the radius moves three to the right and four units up.

This means that the gradient of the radius is four over three.

This means that the gradient of the tangent is negative three quarters.

This means that y is equal to negative three quarters of x add c.

So we can substitute in the values of the the point three, four.

This gives us that four is equal to negative three quarters multiplied by three add C.

We can then solve this equation to tell us that C is equal to 25 over four.

The equation of the tangent is therefore y is equal to negative three quarters x add 25 over four.

Here is a question for you to try.

Pause the video to complete your task and resume once you're finished.

Here is the answer.

Point B is negative four, eight.

This means that the gradient of the radius to point B is negative two.

And the gradient of the tangent is a half.

Here is a question for you to try.

Pause the video to complete your task, and resume once you're finished.

Here is the answer.

This question specifies to leave the equation in the form axe add by add c is equal to zero.

Once you have got y is equal to negative four thirds x add 50 over three, you can multiply all of these terms by three to give three Y is equal to negative four x add 50.

Can then subtract 50 and add four x to both sides to give the final answer in the correct form.

Here is a question for you to try.

Pause the video to complete your task, and resume once you're finished.

Here is the answer.

Once you have found the equations of the two tangents, you're going to solve these as pair of simultaneous equations to find the point of intersection.

That's all for this lesson.

Thanks for watching.