Question

$(2,0)$ is a point on the line joining the points $(2x^2,4x)$ and $(2y^2,4y)$. Then,

• A. $\frac{1}{y}=\frac{-1}{x}$
• B. $\frac{1}{y}=-x$
• C. $x=y$
• D. $x=-y$

Answer 1

The correct option is B

$\frac{1}{y}=-x$

If $(2,0)$ is a point on the line joining the points $(2x^2,4x)$ and $(2y^2,4y)$, then we must have, slope of AB = slope of BC

We know that the slope(m) of the line formed by joining the points $(x_1,y_1)$ and $(x_2,y_2)$ is given by

$m=\frac{y_2-y_1}{x_2-x_1}$

Then, we must have,

$\frac{4x-0}{2x^2-2}=\frac{4y-4x}{2y^2-2x^2}$

$⟹\frac{4x}{2(x^2-1)}=\frac{4(y-x)}{2(y+x)(y-x)}$ [$\because a^2-b^2=(a+b)(a-b)$]

$⟹\frac{x}{x^2-1}=\frac{1}{y+x}$

$⟹x^2-1=x(y+x)$

$⟹x^2-1=xy+x^2$

$⟹xy=-1$