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The Circle — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSThe Circle5 Marks
Question
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.
✓

Answer

Let x2 + y2 + 2gx + 2fy + c = 0 ........(1)
be the required cirde.
Now (1) passes through $\text{P}=(1,\ 1)\&\ \theta(2,\ 2)$
$\therefore$ 1 + 1 + 2g + 2f + C = 0 ........... (2)
4 + 4 + 4g + 4f + c = 0 ............... (3)
Also racius = 1
$\Rightarrow\sqrt{\text{g}^2+\text{f}^2-\text{c}}=1$
$\Rightarrow\text{g}^2+\text{f}^2-\text{c}1\ .......(4)$
from (2) & (4)
$\text{g}+\text{f}+\frac{\text{c}}{2}=-1\ \&$
$\text{g}+\text{f}+\frac{\text{c}}{4}=-2$
on subtraction
and $\text{g}+\text{f}=-3\ ........(4)$
From (4) $\text{g}^2+\text{f}^2=5$ $\big\{\therefore(\text{g}+\text{f})^2=\text{g}^2+\text{f}^2+2\text{g}\text{f}\big\}$
$\therefore2\text{gf}=4$ $\Rightarrow9=5+2\text{gf}$
$\text{gf}=2$
so, $(\text{g}-\text{f})^2=\text{g}+\text{f})^2-4\text{gf}=9-8=1$
$\therefore\text{g}-\text{f}=\pm1\ .........\ (4)$
Solving (5) & (6) we get
$\text{g}=-1$ or $-2\ \&$
$\text{f}=-2$ or $-1$
Thus, required circle
$\text{x}^2+\text{y}^2-2\text{x}-4\text{y}+4=0$
$\text{x}^2+\text{y}^2-4\text{x}-2\text{y}+4=0$

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