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Introduction to LOCI

From point ...

Question

From point $(3, 4)$ , chords are drawn to the circle $x^{2} + y^{2} - 4 x = 0$ . The locus of the midpoints of the chords is

A

x^{2} + y^{2} - 5 x - 4 y + 6 = 0

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B

x^{2} + y^{2} + 5 x - 4 y + 6 = 0

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C

x^{2} + y^{2} - 5 x + 4 y + 6 = 0

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D

x^{2} + y^{2} - 5 x - 4 y - 6 = 0

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Solution

The correct option is A $x^{2} + y^{2} - 5 x - 4 y + 6 = 0$
Centre of the circle $= (2, 0)$

Let the locus be $(h, k)$
Slope of $O M = m_{1}$
Slope of $P M = m_{2}$
$\Rightarrow m_{1} m_{2} = - 1$
$\Rightarrow (\frac{4 - k}{3 - h}) (\frac{k - 0}{h - 2}) = - 1$
$\Rightarrow h^{2} + k^{2} - 5 h - 4 k + 6 = 0$
Hence, locus is $x^{2} + y^{2} - 5 x - 4 y + 6 = 0$

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