Find the equation of the circle passing through the Points (4, 1)and (6, 5) and whose centre is on the line is 4x + y = 16
Answers
Let say the equation 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 eqn (3) and (4), we obtain h = 3 and k = 4.
On substituting the values of h and k in eqn (1), we obtain
(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 + y2² – 8y + 16 = 10
x² + y² – 6x – 8y + 15 = 0
- https://brainly.in/question/34646638
Answer:
x^2 + y^2 – 6x – 8y + 15 = 0
Step-by-step explanation:
Let the equation of the required circle be (x – h)^2 + (y – k)^2 = r^2.
Since the circle passes through points (4, 1) and (6, 5),
(4 – h)^2 + (1 – k)^2 = r^2 …………………. (1)
(6 – h)^2 + (5 – k)^2 = r^2 …………………. (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 + h^2 + 1 – 2k + k^2 = 36 – 12h + h^2 + 25 – 10k + k^2
⇒ 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 = r^2
⇒ (1)^2 + (– 3)^2 = r^2
⇒ 1 + 9 = r^2
⇒ r^2 = 10
⇒ =√10
Thus, equation of the required circle is
(x – 3)^2 + (y – 4)^2 = (√10)^2
x^2 – 6x + 9 + y^2 – 8y + 16 = 10
x^2 + y^2 – 6x – 8y + 15 = 0