Find the equation of point C such that C is a right angle.

Question

A(-1,4) and B(7,5) are two ends of the diameter if a circle.

Answer 1

The angle in a semicircle is a right angle, so C must be on the circumference of the circle.

The average of the endpoints of the diameter give the position of the centre: ((7-1)/2,(4+5)/2)=(3,9/2). The distance between the points gives the diameter=2r, twice the radius.

r=½√((7-(-1)^2)+(5-4)^2)=√65/2. So the equation of the circle is (x-3)^2+(y-9/2)^2=65/4

Multiply through by 4: 4(x-3)^2+(2y-9)^2=65. This can be expanded to 4(x^2-6x+9)+4y^2-36y+81=65.

4x^2+4y^2-24x-36y+36+81-65=0; 4x^2+4y^2-24x-36y+52=0.