Question

Equation of the curve whose gradient at any point (x,y) on it is $\frac{x-a}{y-b}$ and which passes through the origin is:

• A. $x^2-y^2=2(ax-by)$
• B. $x^2+y^2=2(ax+by)$
• C. $x^2-y^2=2(bx+ay)$
• D. $x^2+y^2=2(ax-by)$

Answer 1

The correct option is A $x^2-y^2=2(ax-by)$

$\Rightarrow \frac{dy}{dx}=\frac{x-a}{y-b}$
$\Rightarrow \int (y-b)dy=\int (x-4)dx+c$
$\Rightarrow y^2-2by+b^2=x^2-2ax+a^2+c{a}^2-a^2$
$\Rightarrow (y-b{)}^2=(x-a{)}^2+c$
put (x,y) = ( 0,0)
$\Rightarrow c=b^2-a^2$
$\Rightarrow y^2-2by+b^2=x^2-2ax+a^2+b^2-a^2$
$\Rightarrow x^2-y^2=2(ax-by)$