Question

Let x(x - a) + y(y - 1) be a circle. If two chords from (a, 1) bisected by the x-axis are drawn to the circle then the condition required is

• A. $a^2$ = 8
• B. $a^2$ $<$ 8
• C. $a^2$ $>$ 8
• D. a=0

Answer 1

The correct option is C $a^2$ $>$ 8

The equation of the circle is $x^2$ + $y^2$ -ax -ay = 0
Let (x, 0) be the midpoint of a chord drawn the point (a, 1)
Then the equation of the chord bisected at (x${}_1$, 0) is $x{x}_1-\frac{9}{2}(x+x_1)-\frac{1}{2}(y+0)=x_1^2-a{x}_1$
$\Rightarrow 2x_1^2-2a{x}_1=2x_{1}x-ax-a{x}_1-y.$
The chord passes through (a, 1)
$\Rightarrow 2x_1^2-3a{x}_1+a^2+1=0$
$x_1$ is real $\Rightarrow 9a^2-4\left(2\right)(a^2+1)>0$ $\Rightarrow 9a^2-8as-8>$ 0 $\Rightarrow a^2$ $>$ 8