Q. 2 B4.0( 4 Votes )

# Find the equation of the circle passing through the points :

(1, 2), (3, - 4) and (5, - 6)

Answer :

Given that we need to find the equation of the circle passing through the points (1,2), (3, - 4) and (5, - 6).

We know that the standard form of the equation of a circle is given by:

⇒ x^{2} + y^{2} + 2ax + 2by + c = 0 ..... (1)

Substituting (1,2) in (1), we get

⇒ 1^{2} + 2^{2} + 2a(1) + 2b(2) + c = 0

⇒ 1 + 4 + 2a + 4b + c = 0

⇒ 2a + 4b + c + 5 = 0 ..... (2)

Substituting (3, - 4) in (1), we get

⇒ 3^{2} + (- 4)^{2} + 2a(3) + 2b(- 4) + c = 0

⇒ 9 + 16 + 6a - 8b + c = 0

⇒ 6a - 8b + c + 25 = 0 ..... (3)

Substituting (5, - 6) in (1), we get

⇒ 5^{2} + (- 6)^{2} + 2a(5) + 2b(- 6) + c = 0

⇒ 25 + 36 + 10a - 12b + c = 0

⇒ 10a - 12b + c + 61 = 0 .....(4)

Solving (2), (3), (4) we get

⇒ a = - 11, b = - 2,c = 25.

Substituting these values in (1), we get

⇒ x^{2} + y^{2} + 2(- 11)x + 2(- 2) + 25 = 0

⇒ x^{2} + y^{2} - 22x - 4y + 25 = 0

∴ The equation of the circle is x^{2} + y^{2} - 22x - 4y + 25 = 0.

Rate this question :

The equation of the circle which touches the axes of coordinates and the line and whose centres lie in the first quadrant is x^{2} + y^{2} – 2cx – 2cy + c^{2} = 0, where c is equal to

Find the equation of the circle which circumscribes the triangle formed by the lines:

y = x + 2, 3y = 4x and 2y = 3x

RD Sharma - MathematicsThe equation of the circle concentric with x^{2} + y^{2} – 3x + 4y – c = 0 and passing through (- 1, - 2) is

Find the equation of the circle which passes through (3, - 2), (- 2, 0) and has its centre on the line 2x – y = 3.

RD Sharma - MathematicsShow that the points (5, 5), (6, 4), (- 2, 4) and (7, 1) all lie on a circle, and find its equation, centre, and radius.

RD Sharma - MathematicsIf the circle x^{2} + y^{2} + 2ax + 8y + 16 = 0 touches x - axis, then the value of a is

Find the equation of the circle which passes through the points (3, 7), (5, 5) and has its centre on line x – 4y = 1.

RD Sharma - MathematicsFind the equation of the circle passing through the points :

(5, - 8), (- 2, 9) and (2, 1)

RD Sharma - MathematicsThe circle x^{2} + y^{2} + 2gx + 2 fy + c = 0 does not intersect x - axis, if

Find the equation of the circle passing through the points :

(1, 2), (3, - 4) and (5, - 6)

RD Sharma - Mathematics