ISI2014-DCG-57

Question

If a focal chord of the parabola $y^2=4ax$ cuts it at two distinct points $(x_1,y_1)$ and $(x_2,y_2)$, then

• A. $x_1x_2=a^2$
• B. $y_1y_2=a^2$
• C. $x_1x_2^2=a^2$
• D. $x_1^2x_2=a^2$

Comments

$A$ is the answer.

Answer 1

Answer: $A$

Equation of the chord passing the focal point of the parabola $\mathrm{y^2 = 4ax}$ is $\mathrm{y = m(x-a)}$

$\textrm x-$coordinates of the intersection of this line to the parabola are the solutions of equation.

$\Rightarrow \mathrm {m^2(x-a)^2 = 4ax}$

$\Rightarrow \mathrm {m^2.x^2-2a(m^2+2)x+m^2.a^2 = 0}$

Product $\mathrm {x_1.x_2 = \frac{m^2.a^2}{m^2} = a^2}$

${\therefore \mathbf A}$ is the correct answer.