Find the equation to the circle which passes through the points (… - Vidyadip

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.

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Answer

Let $x^2 + y^2 + 2gx + 2fy + c = 0$ ........(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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