Question

Find the equation of a line passing through the point 4, 1 and 65 and whose centre is on the line 4 x + y is equal to 16

Answer 1

use the formula (x-h)^2+(y-k)^2=r^2

put the value of the point in passing line .

Answer 2

Answer:

Step-by-step explanation:

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

✔️✔ 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)

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

4h + k = 16 …... (3)

✔️✔️From (1) and (2),

(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)

solving (3) and (4),

h = 3

 k = 4.

✔️✔️On substituting the values of h and k in equation (1)

we get......

(4 – 3)2 + (1 – 4)2 = r2

⇒ (1)2 + (– 3)2 = r2

⇒ 1 + 9 = r2

⇒ r2 = 10

⇒r=√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 = 0⭕️

is the required equation