six is the maximum number of common chords of a parabola and a circle

Question

The maximum number of common chords of a parabola & a circle can be?

Answer is 6. But I have no idea How! Please experts explain it in detail

Answer 1

This is because there is a maximum four points of intersection between a parabola and circle. You can show this algebraically:

Say the parabola's equation is $x^2=4ay $ and the circle's equation is $x^2+y^2+2fx+2gy+h=0$

then by substituting $y=x^2/4a$ into the circle's equation, we get:

$\frac{x^4}{16a^2}+x^2+2fx+2gy+h=0$ which is a degree 4 polynomial hence has at max 4 distinct real roots hence 4 different points of intersection (since the parabola equation is many to one).

With 4 points of intersection you can then form a common chord by choosing any two of the four points and joining them; hence you can have at max $4 \choose 2$$=6$ common chords.