Find the equation of a circle, two of whose diameters are along the straight lines x – 2y = 7 and x + 2y + 1 = 0 and which passes through the point (1, 4).
Let AB and EF be the two diameters of the circle and have equations
x - 2y = 7 and x + 2y = -1 respectively.
The centre C of the circle is the point of intersection of the two diameters
∴ For C,
x - 2y = 7 ...(i)
x + 2y = -1 ...(ii)
Adding, we get 2x = 6 x = 3
and 3 - 2y = 7 y = -2
∴ The co-ordinates of centre C of the circle are (3, -2)
Also, the circle passes through point P(1, 4)
∴ Radius of the circle, r = CP =
h = 3, k = -2,
The equation of the circle is:
or
which is the required equation of the circle.