Question

What is the maximum number of intersection points a hyperbola and a circle could have?

Answer 1

The right option is 4.

Answer 2

Answer: A. 4

Step-by-step explanation:

The characteristic equation of a circle is

$x^2+y^2=r^2$

The characteristic equation of a hyperbola is

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

The intersection points will be obtained as the solution of the both characteristic equations.

$x^2+y^2=r^2$.......(1)

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

⇒$y^2=r^2-x^2$.....(from 1)

put this in (2)

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

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

⇒$y^2=r^2-x^2\\x^2(\frac{1}{a^2}+\frac{1}{b^2})=1+\frac{r^2}{b^2}$

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

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

So there are 2 different values of x and two different values of y, thus the maximum number of intersection points  is 4.