Question

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

A. 4
B. 3
C. 1
D. 2 ...?

Answer 1

Answer: A. 4

Explanation:

The characteristic equation of a circle is

`x^2+y^2=r^2`

The characteristic equation of a hyperbola is

`(x^2)/(a^2)-(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)

`(x^2)/(a^2)-(y^2)/(b^2)=1`.............(2)

⇒
`y^2=r^2-x^2`.....(from 1)

put this in (2)

`(x^2)/(a^2)-(r^2-x^2)/(b^2)=1`

⇒
`y^2=r^2-x^2\\(x^2)/(a^2)+(-x^2)/(b^2)=1+(r^2)/(b^2)`

⇒
`y^2=r^2-x^2\\x^2((1)/(a^2)+(1)/(b^2))=1+(r^2)/(b^2)`

⇒
`y^2=r^2-x^2\\x^2((a^2+b^2)/(a^2b^2))=(b^2+r^2)/(b^2)`

`\Rightarrow\ x^2=(a^2(b^2+r^2))/(a^2+b^2)\\y^2=(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.

Answer 2

The right option is 4.