Let and be the endpoints of a diameter of a circle in the Cartesian plane.

MATHEMATICS · Multi-part · A Level · Mathematics · Coordinate geometry in the (x y) plane · Coordinate geometry in the (x y) plane

Problem

Let

P(x_1,y_1)

and

Q(x_2,y_2)

be the endpoints of a diameter of a circle in the Cartesian plane. (i) Show that the midpoint

M\bigl(\tfrac{x_1+x_2}{2},\;\tfrac{y_1+y_2}{2}\bigr)

is equidistant from

P

and

Q

. Hence explain why

M

must be the centre of the circle. (ii) Using the result of part (i), derive the equation of the circle in terms of

x_1,\,y_1,\,x_2

and

y_2

. (iii) Verify your derived equation by substituting the coordinates

P(3,-5)

and

Q(-1,7)

and writing the specific equation of that circle in standard form.