Question

A circle with radius $|a|$ and centre on y-axis slides along it and a variable lines through $(a,0)$ cuts the circle at points $P$ and $Q$. The region in which the point of intersection of tangents to the circle at points $P$ and $Q$ lies is represented by

• A. $y^2\ge 4(ax-a^2)$
• B. $y^2\le 4(ax-a^2)$
• C. $y\ge 4(ax-a^2)$
• D. $y\le 4(ax-a^2)$

Answer 1

The correct option is A $y^2\ge 4(ax-a^2)$

Let the centre be $(0,\alpha )$ equation of circle $x^2+(y-\alpha {)}^2=|a{|}^2$
$\therefore$ Equation of chord of contact for P(h, k) is $xh+yk-\alpha (y+k)+{\alpha }^2-a^2=0$
It passes through $(a,0)$.
$\Rightarrow a^2-\alpha k+ah-a^2=0$
As $\alpha$ is real
$\Rightarrow k^2-4(ah-a^2)\ge 0$