How to write a system of equation that satisfies the conditions "two parabolas that intersect in two points?"
1 Answer
Jan 11, 2017
#{ (y = 2x^2), (y = x^2+1) :}#
Explanation:
We could give the two parabolas the same axis, with their vertices one above the other and make the 'lower' parabola 'steeper' like this:
#{ (y = 2x^2), (y = x^2+1) :}#
These will intersect at
graph{(y-2x^2)(y-x^2-1) = 0 [-4.605, 5.393, -1, 4]}
More generally the following system will satisfy the conditions:
#{ (y = a_1 x^2+b_1 x + c_1), (y = a_2x^2 + b_2x+c_2) :}#
where
#(b_2 - b_1)^2 > 4(a_2 - a_1)(c_2 - c_1)#
These conditions ensure that the quadratics represent parabolas of different 'steepness' and the discriminant of the difference is positive, resulting in two distinct real roots of:
#(a_2-a_1)x^2 + (b_2-b_1)x+(c_2-c_1) = 0#
