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  4. If one end of the diameter is (1, 1) and the other end lies on the line x + y = 3 , then locus of centre of circle is

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Q. If one end of the diameter is $ (1, 1) $ and the other end lies on the line $ x + y = 3 $ , then locus of centre of circle is

MHT CETMHT CET 2008

Solution:

Let the other end be $(t, 3-t)$ So, the equation of the variable circle is
$(x-1)(x-t)+(y-1)(y-3+t)=0 $
$\Rightarrow \,\, x^{2}+y^{2}-(1+t) x-(4-t) y+3=0$
$\therefore \,\,$ The centre $(\alpha, \beta)$ is given by
$\alpha=\frac{1+t}{2}, \beta=\frac{4-t}{2}$
$\Rightarrow \,\,\,2 \alpha+2 \beta=5$
Hence, the locus is $2 x+2 y=5$