Question

"Consider the following conic equations:

7x²−6 √5n x+13y²−51=0
5x²−5√3x+13y² −cs=0
For each equation:
Sketch the graph of the conic section it represents, specifying whether it's an ellipse, hyperbola, parabola, or a degenerate case.
If necessary, find the important properties of the conic section, such as the foci, eccentricity, major and minor axes, and any other relevant features.

Answer 1

The first equation represents an ellipse, and the second equation represents a parabola.

Step-by-step explanation:

The given equations are:

1. 7x²−6√5nx+13y²−51=0

2. 5x²−5√3x+13y²−cs=0

To sketch the graph of each equation and determine the conic section it represents, we need to analyze their forms. The first equation can be rearranged to:

7x² + 13y² − 6 √5n x = 51

Comparing this equation to the standard form of an ellipse, which is x²/a² + y²/b² = 1, we can see that both the terms containing x and y are squared and have positive coefficients. Therefore, the first equation represents an ellipse.

The second equation can be rearranged to:

5x² − 5√3x + 13y² − cs = 0

Comparing this equation to the standard form of a parabola, which is x = ay² + by + c, we can see that it only contains a quadratic term with x and no quadratic term with y. Thus, the second equation represents a parabola.