The points 𝐴 one, negative one; 𝐵 negative one, five; 𝐶 17, 11; and 𝐷 19, five form a rectangle. What is the equation of the circle that contains all four points?
So, the first thing we’ve done is drawn a sketch. And this is gonna help us see what is going on. So, we’ve plotted each of the points on our coordinate axes. And we’ve joined them to form a rectangle. So, in the question, we’re asked to find the equation of the circle that contains all these four points.
So, what we’ve done is drawn a little explosion on the side of the diagram. So, we’ve got our rectangle. And it’s touching the circle, or the circumference of a circle, at each of its four vertices. Now, how can this help us? Well, I’ve also drawn on a line. And this line is the diameter. And we know that it’s going to be the diameter of our circle. So, the diagonal of the rectangle is the diameter of the circle because of one of our circle theorems.
And that’s the circle theorem that tells us that the angles subtended at circumference by a semicircle is 90 degrees. Well, we know that the two angles I’ve shown here are 90 degrees cause they’re right angles because it’s a rectangle. So therefore, the line that subtends them must be the diameter because it must be a semicircle. And then, furthermore, what we know is that the midpoint of this line must be the centre of the circle. So, the midpoint of our diagonal must be the centre of the circle. So, great, we can now use this to solve the problem.
Well, what we need to do is we need to find the equation of the circle. Well, we’ve got a general form for the equation of a circle. That is that 𝑥 minus 𝑎 all squared plus 𝑦 minus 𝑏 all squared is equal to 𝑟 squared, where the centre of a circle is 𝑎, 𝑏 and the radius is 𝑟. So, now, what we can do is we can find our 𝑀 because our 𝑀 is going to be, as we’ve said, the centre of the circle. And we want that because that forms part of our equation.
Well, 𝑀, the midpoint that we’re looking for, is the midpoint of the diagonal. And this diagonal is the line 𝐴𝐶. But what we also have is a general formula we can use to help us find the midpoint of any straight line where we have the two points at either end. So, that general formula is that if we want to find the midpoint of a line, then the 𝑥-coordinate is 𝑥 one plus 𝑥 two divided by two. So, that’s the 𝑥-coordinates of both the points added together and then divided by two. And then, the 𝑦-coordinate is 𝑦 one plus 𝑦 two over two.
So, what we’ve done is we’ve labelled the points. And we’ve got 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. So therefore, if we apply this, what we’re gonna get is 𝑀 is gonna be equal to — well, the 𝑥-coordinate is gonna be one plus 17 over two and the 𝑦-coordinate is going to be negative one plus 11 over two. So therefore, the midpoint of our diagonal, or the centre of our circle, is gonna have the coordinates nine, five.
So, great, now we found the centre of our circle, all we need to do is find the radius. Well, from taking a look at our diagram, we could see that our radius would be half of our diameter. So therefore, it could be represented by 𝐴𝑀 or 𝑀𝐶. So, now, what we can do to find the radius is substitute some values of coordinates we know into the general form of the equation of a circle. And this is gonna help us find out what 𝑟 is going to be.
So therefore, what we’re gonna do is we’re gonna substitute in the 𝑥- and 𝑦-coordinates, so the 𝑥- and 𝑦-values, from 𝐴 into our equation of a circle. So, that’s 𝑥 equals one and 𝑦 equals negative one. So, we’re going to substitute it into the equation of the circle, which is 𝑥 minus 𝑎 all squared plus 𝑦 minus 𝑏 all squared equals 𝑟 squared. And we’ve got one minus nine all squared plus negative one minus five all squared equals 𝑟 squared. And that’s remembering that our 𝑎 and 𝑏 were nine and five, respectively.
So, this is gonna give us negative eight squared plus negative six squared is equal to 𝑟 squared. So, we’re gonna get 64 plus 36 equals 𝑟 squared, which is gonna give us 100 is equal to 𝑟 squared. Great! We don’t actually have to go any further than this because what we want is 𝑟 squared as 𝑟 squared is what’s featured in our equation of a circle. However, if we did want to find the radius, we could. Because what we would do is take the square root of 100, which would give us 10. We could disregard the negative 10 result because we’re looking at a length.
So, now, that we have all the components we need to form our equation, we can put it together to form the equation of our circle. And when we do this, we get 𝑥 minus nine all squared plus 𝑦 minus five all squared is equal to 100. So, this is the equation of the circle that contains all the four points 𝐴, 𝐵, 𝐶, and 𝐷.