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10. The asymptote parallel to the [tex]y[/tex]-axis of the curve [tex]\left(x^2 + y^2\right) x - a y^2 = 0[/tex] is:
To find the asymptote parallel to the y-axis for the curve given by the equation
[tex]\[
(x^2 + y^2) x - a y^2 = 0,
\][/tex]
we can start by rearranging the equation to make it easier to analyze:
[tex]\[
x^3 + x y^2 - a y^2 = 0.
\][/tex]
Here, we want to find the values of \( x \) such that, as \( y \) tends to infinity, the equation describes an asymptote parallel to the y-axis. Essentially, we seek vertical asymptotes where \( x \) tends to some constant value while \( y \) goes to infinity.
To identify such points, we need to analyze how \( x \) behaves as \( y \) grows very large. Suppose \( y \) is very large, we would expect that terms including \( y^2 \) are dominant. Let's re-examine the equation under this assumption:
[tex]\[
x^3 + (x - a)y^2 = 0.
\][/tex]
Dividing through by \( y^2 \):
[tex]\[
\frac{x^3}{y^2} + (x - a) = 0.
\][/tex]
Since \( y \) is very large, the term \(\frac{x^3}{y^2}\) becomes very small and in the limit can be considered approaching zero. Thus, the dominant terms reduce to:
[tex]\[
x - a \approx 0,
\][/tex]
which implies:
[tex]\[
x \approx a.
\][/tex]
Thus, we determined that as \( y \) approaches infinity, \( x \) approaches the constant \( a \), suggesting that \( x = a \) is a vertical asymptote.
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