Question

Two chords are drawn from the point $P(h,k)$ on the circles $x^2+y^2-hx-ky=0.$ If the y-axis divides both the chords in the ratio $1:3,$ then which of the following is/are correct?

• A. $9k^2>16h^2$
• B. $k^2>48h^2$
• C. $9k^2\ge 16h^2$
• D. $4k^2=h^2$

Answer 1

Answer: (A), (B)

$k^2>48h^2$

Case-1: Ratio $=1:3$

$(0,a)=(\frac{h+3x_1}{4},\frac{k+3y_1}{4})$
$\Rightarrow 0=\frac{h+3x_1}{4},a=\frac{k+3y_1}{4}$
$\Rightarrow x_1=-\frac{h}{3},y_1=\frac{4a-k}{3}$

Since, $(x_1,y_1)$ lies on the circle.
$\Rightarrow x_1^2+y_1^2-h{x}_1-k{y}_1=0$
$\Rightarrow \frac{h^2}{9}+\frac{(4a-k{)}^2}{9}+\frac{h^2}{3}+\frac{k^2-4ak}{3}=0$
$\Rightarrow 4a^2-5ak+(h^2+k^2)=0$

Now, $D>0$ for two different chords.
$25k^2-16(h^2+k^2)>0$
$\Rightarrow 9k^2>16h^2$

Case-2: Ratio $=3:1$

$(0,a)=(\frac{3h+x_1}{4},\frac{3k+y_1}{4})$
$\Rightarrow 0=\frac{3h+x_1}{4},a=\frac{3k+y_1}{4}$
$\Rightarrow x_1=-3h,y_1=4a-3k$

$\Rightarrow 9h^2+(4a-3k{)}^2+3h^2+3k^2-4ak=0$
$\Rightarrow 4a^2-7ak+3(h^2+k^2)=0$

Now, $D>0$ for two different chords.
$49k^2-48(h^2+k^2)>0$
$\Rightarrow k^2>48h^2$