1. ## Maximum number of intersections of two quadratics

Hi there,

I am wondering what is the maximum number of points of intersection of 2 quadratics of the form:

$\displaystyle ax^2 + bxy + cy^2 + dx + ey + f = 0$

I think the answer is 4, because expressing y in terms of x has a $\displaystyle \pm$ sign, and then subbing each of these y-values in to the other quadratic leads to 2 solutions each through use of the quadratic formula. Thus 2 x 2 = 4 is the maximum number of intersections.

This also seems intuitive, if we imagine for example a hyperbola being intersected 4 times by an ellipse. I can't picture in my head any 2 quadratics who intersect more than 4 times.

Anyway if someone could please verify or disprove my assertion I'd be very grateful!

2. Originally Posted by r45
Hi there,

I am wondering what is the maximum number of points of intersection of 2 quadratics of the form:

$\displaystyle ax^2 + bxy + cy^2 + dx + ey + f = 0$

I think the answer is 4, because expressing y in terms of x has a $\displaystyle \pm$ sign, and then subbing each of these y-values in to the other quadratic leads to 2 solutions each through use of the quadratic formula. Thus 2 x 2 = 4 is the maximum number of intersections.

This also seems intuitive, if we imagine for example a hyperbola being intersected 4 times by an ellipse. I can't picture in my head any 2 quadratics who intersect more than 4 times.

Anyway if someone could please verify or disprove my assertion I'd be very grateful!
infinite number of points take the same quadratic for the two curves
if the two curves are different you have 4 points

$\displaystyle ax^2 + bxy + cy^2 + dx + ey + f = 0$

if we take two curves say

$\displaystyle a_1x^2 +b_1xy + c_1 y^2 + d_1 x + e_1 y + f_1 = 0$...(1)

$\displaystyle a_2x^2 +b_2xy + c_2 y^2 + d_2 x + e_2 y + f_2 = 0$ ...(2)

rewrite the first one like

$\displaystyle c_1 y^2+ e_1 y +b_1xy + d_1 x +a_1x^2 + f_1 = 0$

$\displaystyle c_1 y^2 + y(e_1 + b_1x) + d_1x + a_1x^2 +f_1 = 0$

solve for y

$\displaystyle y = \frac{-(e_1 + b_1x) \pm \sqrt{(e_1 + b_1x)^2 - 4c_1( d_1x + a_1x^2 +f_1)}}{2c_1}$

now we have to curves from (1)

$\displaystyle y_1 = \frac{-(e_1 + b_1x) + \sqrt{(e_1 + b_1x)^2 - 4c_1( d_1x + a_1x^2 +f_1)}}{2c_1}$

$\displaystyle y_2 = \frac{-(e_1 + b_1x) - \sqrt{(e_1 + b_1x)^2 - 4c_1( d_1x + a_1x^2 +f_1)}}{2c_1}$

and these two curve dose not intersect with each other you can see that

do the same thing for (2) you will have two curves

$\displaystyle y_3 =\frac{-(e_2 + b_2x) + \sqrt{(e_1 + b_1x)^2 - 4c_2( d_2x + a_2x^2 +f_2)}}{2c_2}$

$\displaystyle y_4 = \frac{-(e_2 + b_2x) - \sqrt{(e_2 + b_2x)^2 - 4c_2( d_2x + a_2x^2 +f_2)}}{2c_2}$

and these two never intersect