Question

$P(p,q)$ is a point on a circle passing through the origin and centered at $C(\frac{p}{2},\frac{q}{2})$. If two distinct chords can be drawn from $P$ such that these chords are bisected by the $X-$axis, then:

• A. p²>8q²
• B. p²>16q²
• C. p²<8q²
• D. p²< 16q²

Answer 1

The correct option is A p²>8q²

It can be seen that the given points $P(p,q),C(\frac{p}{2},\frac{q}{2})$ and the origin are collinear which implies that line $OP$ where $O$ is the origin is a diameter of the given circle.
therefore, equation of given circle is $x(x-p)+y(y-q)=0$
i.e., $x^2+y^2-px-qy=0\cdots \left(1\right)$
let $M(a,0)$ be the mid point of a chord $AP$ (see fig)
then, we have $CM\perp AP$
i.e., slope of $CM\times$ slope of $AP=-1$
$\frac{\frac{q}{2}}{\frac{p}{2}-a}\times \frac{q}{p-a}=-1$
i.e.,$q^2+(p-2a)(p-a)=0$
i.e., $2a^2-3pa+p^2+q^2=0\cdots \left(2\right)$
equation $\left(2\right)$ which is quadratic equation in ${}^{\prime }{a}^{\prime }$ shows that there will be two real and distinct values of a if
the discriminant is $>0$
i.e., if $(3p{)}^2-4\times 2(p^2+q^2)>0$
i,e.,if $p^2>8q^2$ which is the desired result.