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.
