# Find the Equation to the Circle Which Passes Through the Points (1, 1) (2, 2) and Whose Radius is 1. Show that There Are Two Such Circles. - Mathematics

Find the equation to the circle which passes through the points (1, 1) (2, 2) and whose radius is 1. Show that there are two such circles.

#### Solution

Let the equation of the required circle be
$x^2 + y^2 + 2gx + 2fy + c = 0$

It passes through (1, 1) and (2, 2).

∴ $2g + 2f + c = - 2$...(1)

And,

$4g + 4f + c = - 8$ ...(2)
From (1) and (2), we have:
$- 2g - 2f = 6 \Rightarrow g + f = - 3$ ...(3)
∴ From (2) and (3), we have:
$c = 4$
$Also, \sqrt{g^2 + f^2 - c} = 1$
$\Rightarrow g^2 + f^2 = 1 + c = 5$
$\Rightarrow \left( g + f \right)^2 - 2gf = 5$
$\Rightarrow gf = 2$
Using (3), we get:
$g = - 2, - 1$
Correspondingly, we have:
$f = - 1, - 2$
Therefore, the required equations of the circles are
$x^2 + y^2 - 4x - 2y + 4 = 0$
$x^2 + y^2 - 2x - 4y + 4 = 0$

Hence, there are two such circles.

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 12 | Page 32