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10-1.Circle and System of Circles
hard
From the origin chords are drawn to the circle ${(x - 1)^2} + {y^2} = 1$. The equation of the locus of the middle points of these chords is
A
${x^2} + {y^2} - 3x = 0$
B
${x^2} + {y^2} - 3y = 0$
C
${x^2} + {y^2} - x = 0$
D
${x^2} + {y^2} - y = 0$
(IIT-1985)
Solution
(c) The given circle is ${x^2} + {y^2} – 2x = 0$.
Let $({x_1},\;{y_1})$ be the middle point of any chord of this circle,
than its equation is ${S_1} = T$.
or $x_1^2 + y_1^2 – 2{x_1} = x{x_1} + y{y_1} – (x + {x_1})$
If it passes through $(0, 0)$ then
$x_1^2 + y_1^2 – 2{x_1} = – {x_1} \Rightarrow x_1^2 + y_1^2 – {x_1} = 0$
Hence the required locus of the given point $({x_1},\;{y_1})$ is ${x^2} + {y^2} – x = 0$.
Standard 11
Mathematics
