Equations of a Circle

Transcript

0:00Introduction

Hello everyone! Okay, a small puzzle to start. Here is a circle: x squared plus y squared, minus six x, minus eight y, plus nine, equals zero. How long a piece of the x axis does it cut off? Is it two units, or six units? By the end, you will find the surprising answer in one line. Today we learn the equations of a circle, and circle problems come up often in JEE papers. So, let's begin.

0:33Centre and radius form

So, every point of a circle is at the same distance, the radius r, from its centre, h comma k, in round brackets. The distance formula, squared, gives this: x minus h, the whole squared, plus y minus k, the whole squared, equals r squared. The small two means squared. Okay? Now a quick test, on a circle from the course. What are the centre and the radius of x minus one, the whole squared, plus y plus two, the whole squared, equals sixteen? Many students say one comma two, with radius sixteen. Oh no, careful, that is the trap! Here, y plus two is y minus, minus two, so k is minus two. And sixteen is r squared, so the radius is four. Centre one comma minus two, radius four. Got it?

1:21General form

Now, exam questions usually open the brackets. Expand, and you get x squared plus y squared, minus two h times x, minus two k times y, plus a constant, h squared plus k squared minus r squared, equals zero. Okay? So the general form of a circle is written like this: x squared plus y squared, plus two g times x, plus two f times y, plus c, equals zero. Right? Now compare the two lines. Two g sits where minus two h was, and two f sits where minus two k was. So tell me, where is the centre? Yes, exactly, the centre is minus g comma minus f: halve the coefficients of x and y, and change their signs. Matching the constants, the radius is the square root of g squared plus f squared minus c, with the tick-shaped square root sign. Clear? Let me ask you this: is every equation of this form a real circle? No, it is not! What sits inside a square root cannot be negative. So a real circle needs g squared plus f squared minus c to be greater than or equal to zero, the sign on screen. Otherwise there is no real circle. With me?

2:31Diameter form

Next, the diameter form. Let the ends of a diameter be x one, y one, and x two, y two; the small numbers below are subscripts. The circle is x minus x one, times x minus x two, plus y minus y one, times y minus y two, equals zero, because the angle in a semicircle is a right angle. Okay? Take the course's example: the circle on the join of one comma two and five comma minus four as diameter. Put them in: x minus one, times x minus five, plus y minus two, times y plus four, equals zero. Right? Now expand the two brackets: x squared minus six x plus five, and y squared plus two y minus eight. Add them, and the arrow, meaning implies, gives x squared plus y squared, minus six x, plus two y, minus three, equals zero, and that's it!

3:25Parametric form

Now, the parametric form. Every point on the circle is h plus r cos theta, comma, k plus r sin theta. Theta, the Greek letter, is the angle the radius to the point makes with the positive x axis. Clear? On our circle from before, the course takes theta as pi over three; pi is another Greek letter, and this is sixty degrees. So the point is one plus four cos pi over three, comma, minus two plus four sin pi over three. With me? Cos sixty degrees is one half, and sin sixty degrees is root three over two. So the point is three comma minus two plus two root three. Got it?

4:02Intercepts on the axes

The last idea is the intercepts: the length a circle cuts off on an axis. On the x axis, y is zero, so the general form becomes x squared, plus two g times x, plus c, equals zero. Its two roots are the x coordinates of the crossing points. Okay? The coefficient of x squared is one, so the gap between the roots is the root of the discriminant, four g squared minus four c. That simplifies to two root of g squared minus c, the intercept on the x axis. Right? In the same way, put x equal to zero, and the intercept on the y axis is two root of f squared minus c. Clear? Now think about this: what happens to the circle if g squared equals c? Correct, the intercept becomes zero. The two crossing points merge into one, so the circle just touches the x axis. The double arrow means if and only if: it touches exactly when g squared equals c.

4:58Example 1: centre and radius, and the sign of c trap

With me? Okay, the first example from the course. Find the centre and radius of x squared plus y squared, minus four x, plus six y, minus twelve, equals zero. Pause the video, and try it first. Done? Okay, let's check. Match it with the general form. Two g is minus four, two f is six, and c is minus twelve. So g is minus two, and f is three. Okay? The centre is minus g comma minus f, so it is two comma minus three. Halve and change the signs, just like the rule. Right? Now for the radius, g squared is four, f squared is nine, and then we take away twelve. Four plus nine minus twelve is one, so the radius is one, right? No, no, wait, look at the sign of c! It is minus twelve, so minus c is plus twelve. The radius is the root of four plus nine plus twelve, root twenty five, which is five. Centre two comma minus three, and radius five, perfect!

5:58Example 2: intercepts on the axes, circle touching x-axis

Now, remember our puzzle from the start? It is the course's harder example. Find the intercepts of x squared plus y squared, minus six x, minus eight y, plus nine, equals zero, on the axes. Pause the video, and try it first. Done? Okay, let's check. Two g is minus six, so g is minus three. Two f is minus eight, so f is minus four. And c is nine. With me? Now for the x axis, we need g squared minus c. What do you get? It is nine minus nine, which is zero! So the intercept on the x axis is zero, and that is our puzzle solved! The circle just touches the x axis, because g squared equals c. Clear? Now the y axis: f squared minus c is sixteen minus nine, which is seven. So the intercept on the y axis is two root seven. Okay? And a quick check: the centre is three comma four, and the radius is the root of nine plus sixteen minus nine, which is four. The centre is four units above the x axis, exactly one radius, so of course it touches!

7:08Quick revision: circle forms, centre, radius, intercepts

So, quick revision. Centre h comma k, radius r: x minus h, the whole squared, plus y minus k, the whole squared, equals r squared. General form: centre minus g comma minus f, radius root of g squared plus f squared minus c, and g squared plus f squared minus c must not be negative. Ends of a diameter given? Use the diameter form on screen. Intercepts: two root of g squared minus c, and two root of f squared minus c, and the circle touches the x axis if and only if g squared equals c. Now go and try the practice problems on this page. All the best!