Question

Show that all the chords of the curve $3x^2+3y^2-2x+4y=0$ which subtend a right angle at the origin are concurrent. Also, find the point of concurrency.

• A. $(-\frac{1}{3},-\frac{2}{3})$
• B. $(\frac{1}{3},-\frac{2}{3})$
• C. $(-\frac{1}{3},\frac{2}{3})$
• D. $(3,-\frac{2}{3})$

Answer 1

Answer: (B)

$(\frac{1}{3},-\frac{2}{3})$

Let the equation of the chord be $y=mx+c$

Now homogenizing the curve
$3x^2+3y^2-2x\left(\frac{y-mx}{c}\right)+4y\left(\frac{y-mx}{c}\right)=0$
$(3c+2m)x^2+(3c+4)y^2-(2-4m)xy=0$
coefficient of $x^2$+coefficient of $y^2=0$
$3c+2m+3c+4=0$
$\Rightarrow 3c+m+2=0$
$\Rightarrow \frac{m}{3}+\frac{2}{3}+c=0$ .......(1)
$\Rightarrow mx-y+c=0$ .......(2)
$\therefore$ chord (2) always passes through $(\frac{1}{3},-\frac{2}{3})$