Q. 7 C5.0( 1 Vote )

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

x + y = 2, 3x - 4y = 6 and x - y = 0

Answer :

Given that we need to find the equation of the circle formed by the lines:

⇒ x + y = 2

⇒ 3x - 4y = 6

⇒ x - y = 0

On solving these lines we get the intersection points A(2,0), B(- 6, - 6), C(1,1)

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 (2,0) in (1), we get

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

⇒ 4 + 4a + c = 0

⇒ 4a + c + 4 = 0 ..... (2)

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

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

⇒ 36 + 36 - 12a - 12b + c = 0

⇒ 12a + 12b - c - 72 = 0 ..... (3)

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

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

⇒ 1 + 1 + 2a + 2b + c = 0

⇒ 2a + 2b + c + 2 = 0 ..... (4)

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

⇒ a = 2, b = 3,c = - 12.

Substituting these values in (1), we get

⇒ x^{2} + y^{2} + 2(2)x + 2(3)y - 12 = 0

⇒ x^{2} + y^{2} + 4x + 6y - 12 = 0

∴ The equation of the circle is x^{2} + y^{2} + 4x + 6y - 12 = 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