# Find the Equation of the Circle Which Circumscribes the Triangle Formed by the Lines 2x + Y − 3 = 0, X + Y − 1 = 0 and 3x + 2y − 5 = 0 - Mathematics

Find the equation of the circle which circumscribes the triangle formed by the lines 2x + y − 3 = 0, x + y − 1 = 0 and 3x + 2y − 5 = 0

#### Solution

In $∆$ABC:
Let AB represent the line 2x + y − 3 = 0.    ...(1)
Let BC represent the line x + y − 1 = 0.      ...(2)
Let CA represent the line 3x + 2y − 5 = 0.  ...(3)

Intersection point of (1) and (3) is (1, 1).
Intersection point of (1) and (2) is (2, −1).
Intersection point of (2) and (3) is (3, −2).

The coordinates of A, B and C are (1, 1), (2, −1) and (3, −2), respectively.

Let the equation of the circumcircle be

$x^2 + y^2 + 2gx + 2fy + c = 0$
It passes through A, B and C.
∴ $2 + 2g + 2f + c = 0$
$5 + 4g - 2f + c = 0$
$13 + 6g - 4f + c = 0$
$\therefore g = \frac{- 13}{2}, f = \frac{- 5}{2}, c = 16$
Hence, the required equation of the circumcircle is
$x^2 + y^2 - 13x - 5y + 16 = 0$
Concept: Circle - Standard Equation of a Circle
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#### APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 24 The circle
Exercise 24.2 | Q 7.2 | Page 32