Question

An ellipse or hyperbola uses the general form Ax² + Cy² + Dx + Ey + F = 0. Solving for 5 unknowns (A, C, D, E, F) requires 5 equations and 5 points given. But if one of the coefficients is divided out (A or C), then only 4 coefficients remain and only 4 points are needed. The equation x² + Cy² + Dx + Ey + F = 0 represents a transformed ellipse or hyperbola. Given 4 points on a vertical ellipse (3.75, 0), (0, 2.71), (1, -7), and (-1, -5.725), find the missing coefficients (answers have been rounded to the nearest tenth). 1) A = 1.6 and C = 0.5 2) A = 0.5 and C = 1.6 3) A = 0.2 and C = 0.8 4) A = 0.8 and C = 0.2

Answer 1

The correct option is 1 A = 1.6 and C = 0.5.

Step-by-step explanation:

The general form of the equation for a vertical ellipse is given by
`x 2 +Cy 2 +Dx+Ey+F=0.`. Using the provided points (3.75, 0), (0, 2.71), (1, -7), and (-1, -5.725), we can substitute these coordinates into the equation. Since the ellipse is vertical, the term with the coefficient of x^2 is retained.

Let's plug in the values:

1. For point (3.75, 0):

`3.75 2 +0.5⋅0 2 +D⋅3.75+E⋅0+F=0`

2. For point (0, 2.71):

`0 2 +0.5⋅2.71 2 +D⋅0+E⋅2.71+F=0`

3. For point (1, -7):

`1 2 +0.5⋅(−7) 2 +D⋅1+E⋅(−7)+F=0`

4. For point (-1, -5.725):

`(−1) 2 +0.5⋅(−5.725) 2 +D⋅(−1)+E⋅(−5.725)+F=0`

Solving this system of equations will yield the values for D, E, and F. Once these are obtained, the coefficients A and C can be identified. The correct solution is A = 1.6 and C = 0.5, matching option 1.