What is the maximum number of points that 2 parabolas can intersect each other. How can I determine the answer as there are infinite parabolas.

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

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A parabola is a curve defined by the general equation y = ax^2 + bx + c or x = ay^2 + by^2 + c. The first equation is that of a parabola that opens upwards or downwards and the second equation is that of a parabola that opens to the right or the left.

Now it is important to note that two parabolas that have the same equation will overlap each other completely and they effectively intersect at an infinite number of points. As this case has been considered, let us move to parabolas that are unique and do not do this.

Consider two parabolas that have the same orientation, let us assume they open along the y-axis. The general equations of these is `y = a_1x^2 + b_1x + c_1` and `y = a_2x^2 + b_2x + c_2` .

To determine the points of intersection of the two, we would have to solve the equation `a_2x^2 + b_2x + c_2 = a_1x^2 + b_1x + c_1`

or `x^2(a_1 - a_2) + x(b_1 - b_2) + (c_1 - c_2) = 0`

This equation could have two real roots as the maximum power of x is 2. The equation may also have one common root or no real roots. This gives the maximum number of points where the parabola intersect as 2.

If we consider two parabola that have perpendicular orientations. The general equation of these is `y = a_1x^2 + b_1x + c_1` and `x = a_2y^2 + a_2y + c_2`

The points of intersection of these can be determined by solving the equation:

`a_2*(a_1x^2 + b_1x + c_1)^2 + b_2*(a_1x^2 + b_1x + c_1) + c_2 = 0`

This is an equation where the highest power of x is 4. It can have a maximum of 4 real roots. When that is the case, the twp parabolas intersect at 4 distinct points.

The maximum number of points of intersection of two distinct parabolas is 4.

pilinghigher's profile pic

pilinghigher | Student, Doctorate | eNotes Newbie

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If there isn't the constraint that the parabolas are distinct, then the answer is that two parabolas may intersect one another infinitely.  However, since the question likely meant to imply two distinct parabolas, "Justaguide's" solution is likely considered correct for you.

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