Question

$(3,0)$ is a point on the line joining the points $(3x^2,6x)$ and $(3y^2,6y)$. Then,

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

Answer 1

The correct option is C

$xy=-1$

If $(3,0)$ is a point on the line joining the points $(3x^2,6x)$ and $(3y^2,6y)$, 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,

$\frac{6x-0}{3x^2-3}=\frac{6y-6x}{3y^2-3x^2}$

$⟹\frac{6x}{3(x^2-1)}=\frac{6(y-x)}{3(y+x)(y-x)}$ [$\because a^2-b^2=(a+b)(a-b)$]

$⟹\frac{x}{x^2-1}=\frac{1}{y+x}$

$⟹x^2-1=x(y+x)$

$⟹x^2-1=xy+x^2$

$⟹xy=-1$