Question

From $\left(3,4\right)$ chords are drawn to the circle $x^2+y^2-4x=0$. The locus of the mid points of the

• A. $x^2+y^2-5x-4y+6=0$
• B. $x^2+y^2+5x-4y+6=0$
• C. $x^2+y^2-5x+4y+6=0$
• D. $x^2+y^2-5x-4y-6=0$

Answer 1

The correct option is A $x^2+y^2-5x-4y+6=0$

$x^2+y^2-4x=0$

So,$(x-2{)}^2+y^2=0$ has a centre $(2,0)$ and Radius$=2$

Let locus of midpoint of chord passing through $(3,4)$ be $(h,k)$ then,

chord is $\perp$ to the line joining centre & mid point that means

$m_1{m}_2=-1$

$\frac{(4-k)}{(3-h)}\times \frac{\left(k\right)}{(h-2)}=-1$

$4k-k^2=(h-3)(h-2)$

$h^2-5h+6+k^2-4k=0$

$h^2+k^2-5h-4k+6=0$

Put $h=x,k=y$

$x^2+y^2-5x-4y+6=0$

A is correct.