Advertisement Remove all ads

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.

Advertisement Remove all ads

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
  Is there an error in this question or solution?
Advertisement Remove all ads

APPEARS IN

RD Sharma Class 11 Mathematics Textbook
Chapter 24 The circle
Exercise 24.2 | Q 12 | Page 32
Advertisement Remove all ads
Advertisement Remove all ads
Share
Notifications

View all notifications


      Forgot password?
View in app×