The given equation is \( x^2 = -6y \), which is a parabolic equation. The equation can be compared to the standard form of a vertical parabola \( x^2 = 4py \), where \( p \) represents the distance between the vertex and the focus.

To rewrite \( x^2 = -6y \) in this standard form:

\[
x^2 = 4py
\]
Comparing \( x^2 = -6y \) to \( x^2 = 4py \), we can see that \( 4p = -6 \). Solving for \( p \):

\[
p = \frac{-6}{4} = -\frac{3}{2}
\]

Since \( p = -\frac{3}{2} \), the focus is \( \left(0, -\frac{3}{2}\right) \) from the vertex, which is at the origin \( (0, 0) \). Therefore, the **focus** of the parabola is at:

\[
\boxed{(0, -\frac{3}{2})}
\]

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you find the vertex of a parabola from its equation?
2. What are the directrix and axis of symmetry for this parabola?
3. How do you determine whether a parabola opens upwards, downwards, or sideways?
4. What is the equation for a parabola given its focus and vertex?
5. How do we derive the standard form of a parabola from a general quadratic equation?

**Tip:** The sign of \( p \) in a parabola determines whether it opens upwards (positive \( p \)) or downwards (negative \( p \)).

This page provides a step-by-step solution for finding the focus of the parabola given by the equation x^2 = -6y. Learn how to rewrite the equation in standard form and calculate the focus, suitable for students in Grades 9-11 studying conic sections.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation