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. $x y=10$
• B. $x^2=10 y$
• C. $y^2=10 x$
• 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 10 m = \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 x y-4 y+10=0 $
$\text { taking } -\operatorname{sign} \frac{-10}{y}=-x+C \Rightarrow-2=-2+C \Rightarrow C=0 \Rightarrow x y=10 $