Question

The equation of the curve passing through $(2,5)$ and having the area of triangle formed by the $x$-axis the ordinate of a point on the curve and the tangent at the point 5 square units is

• A. $xy=10$
• B. $x^2=10y$
• C. $y^2=10x$
• D. $x^{2}y=10$

Answer 1

Answer: (A)

$\tan \theta =m\Rightarrow \frac{y}{AB}=m\Rightarrow AB=\frac{y}{m}$
$\frac{1}{2}\cdot \frac{y}{m}\cdot y=\pm 5$
$\Rightarrow 10m=\pm y^2$
$10\frac{dy}{dx}=\pm y^2\Rightarrow 10\frac{dy}{y^2}=\pm dx$
$\text{ Integral, }10\left(\frac{-1}{y}\right)=\pm x+C$
$\text{ taking }+\text{ sign }\Rightarrow \frac{-10}{y}=x+C\text{, put }(2,5)\Rightarrow -2=2+C\Rightarrow C=-4$
$\Rightarrow \frac{-10}{y}=x-4\Rightarrow y(x-4)=-10\Rightarrow xy-4y+10=0$
$\text{ taking }-\mathrm{sign}\frac{-10}{y}=-x+C\Rightarrow -2=-2+C\Rightarrow C=0\Rightarrow xy=10$