Find the equation of an equilateral hyperbola passing through a point of a circumference

Question

Let's say I have a circumference with the following equation $x^2+y^2-10=0$, the coordinates of its center are $(0;0)$ and its radius is $\sqrt{10}$.

I need to find the equation a equilateral hyperbola passing through the point A of the circumference of abscissa 1 and positive ordinate.

I know that the equation of the equilateral hyperbola is $$xy = c$$

However I can't still figure out how to find its equation. Perhaps I need to put the equation of the circumference and the one of the hyperbola into a system and then solve it?

Answer 1

In general, you need to find $c$ such that on point $A=(x_a,y_a)$ you have $$x_a^2+y_a^2-r^2=0$$ and $$ x_a y_a = c$$

Set $$y_a =\pm \sqrt{r^2-x_a^2}$$ in the second equation to get $$c = x_a \sqrt{r^2-x_a^2}$$

So your hyperbola is now $x y = x_a \sqrt{r^2-x_a}$.

Answer 2

First, the point A has coordinates (1,3) because it lies on the circumference and has a positive ordinate: $1^2+y^2=10\Rightarrow y=+\sqrt{10-1}=3$. Then you want your equilateral hyperbola to pass through the point $A(1,3)$. Therefore you plug the coordinates of A in the equilateral hyperbola equation $xy=c$ and find $1.3=c\Rightarrow c=3$.