Question

# Asymptotes of the curve $y + a = 0$, $xy\left( {x + y} \right) + a\left( {{y^2} + xy - {x^2}} \right) - 10{a^2}y = 0$is /are A.$x + a = 0$B.$y - a = 0$C.$x - a = 0$D.$y + a = 0$

Hint:
First we try to understand what is meant by asymptote. Asymptote is the line whose distance is continuously decreasing with a curve but never becomes zero.
If we want to find the condition for x –axis and y –axis as asymptotes then we use below format.
For asymptotes parallel to the x-axis equate the coefficient of higher power of x is zero.
For asymptotes parallel to y-axis equate the coefficient of higher power of y is zero.
First simplify the given algebraic equation and make the coefficient of higher power equal to zero.

Complete step by step solution:
$xy\left( {x + y} \right) + a\left( {{y^2} + xy - {x^2}} \right) - 10{a^2}y = 0$
$\begin{array}{l} y{x^2} + x{y^2} + a{y^2} + axy - a{x^2} - 10{a^2}y = 0\\ \left( {y - a} \right){x^2} + \left( {x - a} \right){x^2} + axy - 10{a^2}y = 0 \end{array}$
Equate the coefficient of higher power of x is zero
$y - a = 0$
Equate the coefficient of higher power of y is zero
$x - a = 0$
option B and c are correct.

Note:
Asymptote of algebraic curve or oblique asymptote : An asymptote which is not parallel to y-axis is called an oblique asymptote.
Let be an asymptote of then
$m = {\lim _{x \to \infty }}\dfrac{y}{x}\,\,and\,\,c = \mathop {\lim }\limits_{x \to \infty } \left( {y - mx} \right)$ suppose is an asymptote of the curve. Put in the equation of the curve and arrange it in descending powers of x, equate to zero the coefficient of two highest degree terms.