Question

The curve such that the intercept on the $x-$axis cut off ti between the origin and the tangent at a point is twice the abscissa and which passes through the point $(1,2)$. If the ordinate of the point on the curve is $\frac{1}{3}$, then the value of abscissa is ____

Answer 1

Let $y=f\left(x\right)$ be the equation of the curve. Let $P(x_1,y_1)$ be the point on the curve.

Equation of tangent at $P(x_1,y_1)$ with slope $m$ is

$y-y_1=m(x-x_1).........\left(1\right)$

Substitute $y_1=0$ we get

$x_1=x-\frac{y}{m}=2\times \text{abcissa}=2\times xm=\frac{dy}{dx}x-\frac{y}{m}=2x\frac{y}{m}=-x\frac{dy}{y}=-\frac{dx}{x}$

Integrating on both sides

$\int \frac{dy}{y}=-\int \frac{dx}{x}\ln y=-\ln x+\ln c\ln xy=\ln cxy=c$

The curve passes through $(1,2)$ therefore $c=2$

ordinate of the curve = $\frac{1}{3}$

$x\times \frac{1}{3}=2x=6$