Name: Equation of a Circle and Position of a Point
Uploaded: 2019-07-19
Duration: 7 min 54 s
Description: Suppose there is a circle and a point, and the position of the point is to be specified. But, you cannot see the circle and the point. Can you find the posit...

Question

Find the equation of the circle which is circumscribed about the triangle whose vertices are $A (- 2, 3), B (5, 2)$ and $C (6, - 1)$ . Find the centre and radius.

Accepted Answer

Let the required equation of the circle be $x^{2} + y^{2} + 2 g x + 2 f y + c = 0 \to (1)$ and its Centre $= (- g, - f)$ , Radius $= \sqrt{g^{2} + f^{2} - c}$ .

Since it passes through each of the points $A (- 2, 3), B (5, 2)$ and $C (6, - 1)$ , each one of these points must satisfy $(1)$ .

$∴ 4 + 9 - 4 g + 6 f + c = 0 \Rightarrow 13 - 4 g + 6 f + c = 0 \to (2)$

$∴ 25 + 4 + 10 g + 4 f + c = 0 \Rightarrow 29 + 10 g + 4 f + c = 0 \to (3)$

$∴ 36 + 1 + 12 g - 2 f + c = 0 \Rightarrow 37 + 12 g - 2 f + c = 0 \to (4)$

Now, equation $(3)$ $-$ equation $(2)$ , then we get: $16 + 14 g - 2 f = 0 \Rightarrow 8 + 7 g - f = 0 \to (5)$

And also, equation $(4) -$ equation $(3)$ , then we get: $8 + 2 g - 6 f = 0 \Rightarrow 4 + g - 3 f = 0 \to (6)$

Now, equation $(5) \times 3 -$ equation $(6)$ , then we get: $20 + 20 g = 0 \Rightarrow g = - 1$

Substitute $g = - 1$ in equation $(5)$ , then we get: $8 - 7 - f = 0 \Rightarrow f = 1$

Now, substitute $f = 1, g = - 1$ in equation $(2)$ , then we get: $13 + 4 + 6 + c = 0 \Rightarrow c = - 23$

Hence, the required equation of the circle is $x^{2} + y^{2} - 2 x + 2 y - 23 = 0$

Thus, Centre $(- g, - f) = (1, - 1)$ and Radius $\sqrt{g^{2} + f^{2} - c} = \sqrt{1 + 1 + 23} = 5$ units.

Find the equation of the circle which is circumscribed about the triangle whose vertices are $A (- 2, 3), B (5, 2)$ and $C (6, - 1)$ . Find the centre and radius.

Important Questions on Circle

Learn and Prepare for any exam you want !

Find the equation of the circle which is circumscribed about the triangle whose vertices are A(-2,3), B(5,2) and C(6,-1). Find the centre and radius.

Important Questions on Circle

Learn and Prepare for any exam you want !

Find the equation of the circle which is circumscribed about the triangle whose vertices are $A (- 2, 3), B (5, 2)$ and $C (6, - 1)$ . Find the centre and radius.