Question

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4x + y = 16.

Answer 1

Let the equation of the required circle be (x – h)² + (y – k)² = r².

Since the circle passes through points (4, 1) and (6, 5),

(4 – h)² + (1 – k)² = r²   …(1)

(6 – h)² + (5 – k)² = r²   …(2)

Since the centre (h, k) of the circle lies on line 4x + y = 16,

4h + k = 16 … (3)

From equations (1) and (2), we obtain

(4 – h)² + (1 – k)² = (6 – h)² + (5 – k)²

⇒ 16 – 8h + h² + 1 – 2k + k² = 36 – 12h + h² + 25 – 10k + k²

⇒ 16 – 8h + 1 – 2k = 36 – 12h + 25 – 10k

⇒ 4h + 8k = 44

⇒ h + 2k = 11 … (4)

On solving equations (3) and (4), we obtain h = 3 and k = 4.

On substituting the values of h and k in equation (1), we obtain

(4 – 3)² + (1 – 4)² = r²

⇒ (1)² + (– 3)² = r²

⇒ 1 + 9 = r²

⇒ r² = 10

`=> r  = sqrt10`

Thus, the equation of the required circle is

(x – 3)² + (y – 4)² = `(sqrt(10))^2`

x² – 6x + 9 + y² ­– 8y + 16 = 10

x² + y² – 6x – 8y + 15 = 0