Find equation of a parabola when the area cut by a straight line and two intersection points are given.

Question

1. I have equation of a straight line $y = -\frac{0.45}{0.248}x - 1.1330645$
   
2. It intersects a quadratic function graph (parabloa) at two points (-0.652, 0.05) & (-0.9, 0.5)
   
3. I have the area bounded between them, it is 0.19.
   I need to know the equation of the quadratic function.
   This is what i tried,
   
   $\int_{0.05}^{0.5}[{(-\frac{y+1.330645}{1.81451613}) - (ay^2+by+c)] dy=0.19}$

Answer 1

You have the line $ y = m x + b $ and two points $P = (P_x, P_y)$ and $Q = (Q_x, Q_y )$ on the parabola $ x = A y^2 + B y + C $ where $A,B,C$ are unknown, but it is assumed that $A \gt 0$.

The equation of the line can be written as $ x = \dfrac{1}{m}(y - b)$

The area is given as $A_0$ where

$A_0 = \displaystyle \int_{P_y}^{Q_y} \dfrac{1}{m}(y - b) - (A y^2 + B y + C) dy \\ = \dfrac{1}{ m} \left( \dfrac{1}{2} (Q_y^2 - P_y^2) - b (Q_y - P_y) \right) - \dfrac{1}{3} A (Q_y^3 - P_y^3) + \dfrac{1}{2} B (Q_y^2 - P_y^2) + C (Q_y - P_y) $

In addition, we have

$ A P_y^2 + B P_y + C = P_x $

$ A Q_y^2 + B Q_y + C = Q_x $

Now it is a matter of substituting the given values for $m,b, P_x, P_y, Q_x, Q_y$ and solving the resulting $3 \times 3$ linear system for $A,B,C$.

Answer 2

THIS IS NOT AN ANSWER, but I can't post picture in the comment, and my clarification request is long.

You are not providing enough information. Look at the black line below $y=-x-m$, and the intersections with the red, green, and blue curves. The RGB curves all pass through two points on the line, and all are parabolas. The green one is the form $y=ax^2+bx+c$, with the symmetry axis vertical. The blue one is the form $x=ay^2+by+c$, with the symmetry axis horizontal. The red one is the form $$x+y=a(x-y)^2+b(x-y)+c$$ I choose this form, so that it's axis of symmetry is perpendicular to the original straight line. All colored curves are parabolas, and the parameters can be adjusted so that they will pass through the given points and the area is the given number. In principle there are an infinite numbers of parabolas that obey your given conditions. If you look at the picture above, I could draw the symmetric curves with respect to the straight line.

In order for your problem to have a unique solution, you need to specify the axis of symmetry and the position of the vertex with respect too the straight line. In your case, it seems like the axis of symmetry is perpendicular to the straight line in the middle of the two given points, and the vertex is below.