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Show that the Points (5, 5), (6, 4), (−2, 4) and (7, 1) All Lie on a Circle, and Find Its Equation, Centre and Radius.

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Question

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

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Solution

Let the required equation of the circle be

\[x^2 + y^2 + 2gx + 2fy + c = 0\] ...(1)
It is given that the circle passes through (5, 5), (6, 4), (−2, 4).
∴ \[50 + 10g + 10f + c = 0\] ...(2)
\[52 + 12g + 8f + c = 0\] ...(3)
\[20 - 4g + 8f + c = 0\] ...(4)
Solving (2), (3) and (4):
\[g = - 2, f = - 1, c = 20\]
Thus, the equation of the circle is
\[x^2 + y^2 - 4x - 2y - 20 = 0\]  ...(5)
We see that the point (7, 1) satisfies equation (5).
Hence, the points (5, 5), (6, 4), (−2, 4) and (7, 1) lie on the circle.
Also, centre of the required circle = \[\left( 2, 1 \right)\]
Radius of the required circle = \[\sqrt{4 + 1 + 20} = 5\]
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Advanced Concept of Circle - Standard Equation of a Circle
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Chapter 24: The circle - Exercise 24.2 [Page 32]

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R.D. Sharma Mathematics [English] Class 11
Chapter 24 The circle
Exercise 24.2 | Q 6 | Page 32

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