Question

Chords of the parabola $y^2+4y=\frac{4}{3}x-\frac{16}{3}$ which subtend right angle at the vertex pass through

• A. $(\frac{7}{3},-2)$
• B. $(\frac{1}{3},0)$
• C. $(\frac{4}{3},0)$
• D. $(0,\frac{4}{3})$

Answer 1

Answer: (A)

$(\frac{7}{3},-2)$

Equation of the parabola may be written as
${(y+2)}^2=\frac{4}{3}(x-1)$
or $Y^2=4aX$ where $Y=y+2$, $a=\frac{1}{3}$, $X=x-1$
Equation of the chord of this parabola
Joining $P(a{t}_1^2,2a{t}_1)$ and $Q(a{t}_2^2,2a{t}_2)$ is
$Y(t_1+t_2)=2X+2a{t}_1{t}_2$
PQ subtends a right angle at the vertex $(X,Y)=(0,0)$
so $\frac{2a{t}_1}{a{t}_1^2}\times \frac{2a{t}_2}{a{t}_2^2}=-1$
$\Rightarrow$ $t_1{t}_2=-4$
so the equation of the chord PQ is $Y(t_1+t_2)=2(X-4a)$
which passes through $X=4a,Y=0$
$\Rightarrow$ $x-1=4\times 1/3$, $y+2=0$
$\Rightarrow$ $x=1+4/3=7/3$, $y=-2$