Question

10. 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.
11. Find the equation of the circle passing through the​

Answer 1

Step-by-step explanation:

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 ⇒ =√10 Thus, the equation of the required circle is (x – 3)2 + (y – 4)2 = (√10)2 x2 – 6x + 9 + y2 – 8y + 16 = 10 x2 + y2 – 6x – 8y + 15 = 0Read more on Sarthaks.com - https://www.sarthaks.com/33133/find-the-equation-the-circle-passing-through-the-points-and-and-whose-centre-is-on-the-line