Answer to Question #100087 in Analytic Geometry for Tyntyn Evans

Question

A circle and a hyperbola can have a maximum of how many intersection?

Answer 1

We need to find the maximum number of intersections of a circle and a hyperbola.

The Equation of a circle is

$$x^2 + y^2 = r^2 ...……......…..(1)$$

The equation of a hyperbola is

$$\frac {x^2}{a^2} - \frac {y^2}{b^2} =1 ………...……….(2)$$

The intersection points can be obtained as the solution of both the above equations.

From equation (1),

$$y^2 = r^2 - x^2$$

Plug this in equation (2),

$$\frac {x^2}{a^2} - \frac {r^2 - x^2}{b^2} =1$$

$$b^2 x^2 - a^2 r^2 + a^2 x^2 = a^2 b^2$$

$$x^2 ( a^2 + b^2 ) = a^2 b^2 + a^2 r^2$$

$$x^2 = \frac {a^2 (b^2 + r^2)}{a^2 + b^2}$$

$$x = ± \sqrt {\frac {a^2 (b^2 +r^2)} {a^2 + b^2}}$$

Plug the $x^2 = \frac {a^2 (b^2 + r^2)}{a^2 + b^2}$ in $y^2 = r^2 - x^2$

$$y^2 = r^2 - \frac {a^2 (b^2 + r^2) }{a^2 + b^2 }$$

$$y^2 = \frac {a^2 r^2 + b^2 r^2 - a^2 b^2 - a^2 r^2}{a^2 + b^2}$$

$$y^2 = \frac {b^2 (r^2 - a^2)}{a^2 + b^2}$$

$$y = ± \sqrt {\frac {b^2 (r^2 - a^2 )} {a^2 + b^2}}$$

So, the intersecting points are pairs (x,y) taking the sign into account just like (+, + ), (+, -), (-, + ), ( -, -).

The maximum number of intersections is 4.

Answer: the maximum number of intersecting points is 4.