Question

The locus of midpoints of chords of $y^2=4ax$ which subtend a constant angle $\alpha$ at the vertex is

• A.
• B.
• C.
• D. none

Answer 1

The correct option is A

Let $p(x_1,y_1)$ be a point in the locus.
Equation of the chord having $(x_1,y_1)$ as mid point is $y{y}_1-2ax=y_1^2-2a{x}_1\Rightarrow \frac{y{y}_1-2ax}{y_1^2-2a{x}_1}$
Homogeinising $y^2=4ax\left[\frac{y{y}_1-2ax}{y_1^2-2a{x}_1}\right]\Rightarrow (y_1^2-2a{x}_1)y^2-4ax(y{y}_1-2ax)$
$\Rightarrow 8a^2{x}^2-\left(4a{y}_1\right)xy+(y_1^2-2a{x}_1)y^2=0$
$\alpha$ is the angle between the lines $\Rightarrow tan\alpha =\frac{2\sqrt{(2a{y}_1{)}^2-8a^2(y_1^2-2a{x}_1)}}{8a^2+y_1^2-2a{x}_1}$
$\Rightarrow (y_1^2-2a{x}_1+8a^2)ta{n}^2\alpha =16a^2(4a{x}_1-y_1^2)$
$Locusof(x_1,y_1)is(y^2-2ax+8a^2{)}^2=16a^2(4ax-y^2)co{t}^2\alpha$