The problem asks us to identify the type of conic section represented by the given equation: $-\frac{x^2}{9} = \frac{y}{4}$ and $10x^2 - 25y^2 = 100$

1. Problem Description

The problem asks us to identify the type of conic section represented by the given equation:

$-\frac{x^2}{9} = \frac{y}{4}$

and

$10x^2 - 25y^2 = 100$

2. Solution Steps

For equation 6, we have:

$-\frac{x^2}{9} = \frac{y}{4}$

Multiplying both sides by $-36$ gives:

$4x^2 = -9y$

$x^2 = -\frac{9}{4}y$

This equation is of the form $x^2 = 4py$, which is a parabola. Since the coefficient of $y$ is negative, the parabola opens downwards.

For equation 15, we have:

$10x^2 - 25y^2 = 100$

Dividing by 100 gives:

$\frac{10x^2}{100} - \frac{25y^2}{100} = 1$

$\frac{x^2}{10} - \frac{y^2}{4} = 1$

This is the equation of a hyperbola in the form $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, where $a^2 = 10$ and $b^2 = 4$. Since the $x^2$ term is positive, this is a horizontal hyperbola.

3. Final Answer

For equation 6: Parabola.

For equation 15: Horizontal Hyperbola.