if 9(2x+y+2)^2=-4(x-2y+1) represents a parabola, find length of latus rectum

To find the length of the latus rectum of a parabola given by the equation \(9(2x+y+2)^2 = -4(x-2y+1)\), we first need to rewrite the equation in the standard form of a parabola. 

The standard forms of parabolas are:
- \((x - h)^2 = 4p(y - k)\) for a parabola that opens upwards or downwards.
- \((y - k)^2 = 4p(x - h)\) for a parabola that opens sideways.

Let's start by expanding and simplifying the given equation \(9(2x+y+2)^2 = -4(x-2y+1)\):

1. **Expand and simplify**:

   \[ 9(2x + y + 2)^2 = -4(x - 2y + 1) \]

   First, expand the left side:

   \[ 9(4x^2 + 4xy + y^2 + 8x + 4y + 4) = -4(x - 2y + 1) \]

   \[ 36x^2 + 36xy + 9y^2 + 72x + 36y + 36 = -4x + 8y - 4 \]

2. **Combine like terms**:

   \[ 36x^2 + 36xy + 9y^2 + 76x + 28y + 40 = 0 \]

Now, it's challenging to directly compare this form to a standard form of a parabola. Hence, we should look for an alternative way to identify the length of the latus rectum. 

The latus rectum of a parabola \((x-h)^2 = 4p(y-k)\) or \((y-k)^2 = 4p(x-h)\) has length \(|4p|\).

Given the complexity of the algebraic manipulation to get the equation into a standard form directly, we observe that the given quadratic form does represent a conic section and, due to the square and the form of its coefficients, it must be representing a parabola, and we extract the coefficient in relation to the quadratic terms.

For the general form of the quadratic equation \(Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0\) where:
- If \( B^2 - 4AC = 0 \), then it represents a parabola.

Here \(A = 36\), \(B = 36\), and \(C = 9\):

\[ B^2 - 4AC = 36^2 - 4(36)(9) = 1296 - 1296 = 0 \]

Thus, it indeed represents a parabola. 

From the standard form \((2x + y + 2)^2 = -\frac{4}{9}(x - 2y + 1)\), we see that the coefficient of the squared term \( (2x + y + 2)^2 \) should match the general form \(4p\).

So, \(4p = \frac{4}{9} \Rightarrow p = \frac{1}{9}\).

Therefore, the length of the latus rectum \(|4p|\) is \(\frac{4}{9}\).

**The length of the latus rectum of the given parabola is \(\frac{4}{9}\)**.

Would you like any further details or have any other questions?

Here are some questions you might ask next:
1. How to identify the vertex of the parabola?
2. How to determine the direction in which the parabola opens?
3. What is the focus of the parabola?
4. How to convert a quadratic equation to the standard form?
5. How to find the equation of the directrix of a parabola?

**Tip:** When converting quadratic equations to standard forms, always check if you can complete the square to simplify the process.

Learn how to find the length of the latus rectum of a parabola represented by the equation 9(2x+y+2)^2 = -4(x-2y+1). This problem involves understanding the standard form of a parabola, identifying the latus rectum, and applying the discriminant of a quadratic equation. Suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation