Maths Question Can You Solve It?

Question

C2 Ex 4f Q16

The points A (-5,5) B (1,5) C (3,3) D (3,-3) lie on a circle. Find the equation of the circle.

I like to know HOW TO GET the answer, i have no idea what to do, thax.

Answer 1

NO algebra is needed to SOLVE this problem; it's only needed to EXPRESS the solution in equation form!

Take geometrical advantage of the given information. First make (even) a CRUDE sketch of the positions of these points. Then note the following:

The points C and D are "mirror images" with respect to the x-axis. Since the centre of the circle lies on the bisector of all chords, the circle's centre lies on the x-axis, i.e. its y-value is 0. Similarly, the points A and B are both at y = 5, so AB is parallel to the x-axis. Then a perpendicular line through its midpoint will also contain the centre. The x-value of the midpoint of the chord AB is (- 2), so that is the centre's x-value.

Thus the centre is at (- 2, 0).

One more thing you need is the square of the radius. By Pythagoras's Theorem applied between the centre and any of the given points A, B, C or D that you choose, this is 5^2 + 3^2 = 25 + 9 = 34.

So the equation of this circle is:

(x + 2)^2 + y^2 = 34, i.e. x^2 + 4 x + 4 + y^2 = 34, that is:

x^2 + y^2 + 4 x = 30.

CHECK. Lets' check that this does in fact go through (3, 3) :

9 + 9 + 12 = 30.

(It works for all the others, too.)

Live long and prosper.

Answer 2

There is a center of the circle, call it P = (x,y). Also a radius r.

You know the 4 points are on the circle, so

|P - A|^2 = r^2, or
(x+5)^2 + (y-5)^2 = r^2.

Similarly
(x-1)^2 + (y-5)^2 = r^2
(x-3)^2 + (y-3)^2 = r^2
(x-3)^2 + (y+5)^2 = r^2

So you solve from here. As a hint, using the last two equations you can solve for y.