Question

A curve is such that the length of the intercept on the $x-$axis of the tangent at a point is twice the abscissa and passes through the point $(1,2).$ Then the equation of the curve is:

• A. $y^3=8x$
• B. $y^3=-8x$
• C. $xy=2$
• D. $xy=-2$

Answer 1

Answer: (A), (C)

$xy=2$

The equation of the tangent at any point $p(x,y)$ is
$Y-y=\frac{dy}{dx}(X-x)$
Given that length of intercept on $X-$axis$=2$ $(x-$co-ordinates of $p)$
$\Rightarrow |x-y\frac{dx}{dy}|=2x\Rightarrow \frac{dy}{dx}=\frac{-y}{x}\text{(or)}\frac{dy}{dx}=\frac{y}{3x}$
$\Rightarrow -\frac{dy}{y}=\frac{dx}{x}\text{(or)}3\frac{dy}{y}=\frac{dx}{x}$
On integrating we get
$xy=c\text{(or)}3\ln |y|=\ln |x|+\ln |k|$
Since, the curve passes through $(1,2),$
$\Rightarrow c=2,k=8,-8\text{(rejected)}$
Hence, the equation of the required curve is $xy=2$ or $y^3=8x$