Question

The equation of the curve which passes through the point $(2 a, a)$ and for which the sum of the cartesian sub tangent and the abscissa is equal to the constant $a$, is

• A. $y(x-a)=a^{2}$
• B. $y(x+a)=a^{2}$
• C. $x(y-a)=a^{2}$
• D. $x(y+a)=a^{2}$

Answer 1

Answer: (A)

We have,
Cartesian subtangent + abscissa = constant
$\Rightarrow \frac{y}{d y / d x}+x=a $
$\Rightarrow y \frac{d x}{d y}+x=a$
$\Rightarrow \frac{d y}{y}=\frac{d x}{a-x}$
Integrating, we get
$\log y+\log (x-a)=\log c$
$\therefore y(x-a)=c$
As the curve passes through the point $(2 a, a)$, we have $c=a^{2}$
Hence, the required curve is $y(x-a)=a^{2}$.