Find the eccentricity for 2X² + 5Y = 10.
Understand the Problem
The question is asking for the eccentricity of a conic section represented by the equation 2X² + 5Y = 10. To solve this, we will first rewrite the equation in a standard form to identify the type of conic section, and then we will calculate its eccentricity.
Answer
The eccentricity of the conic section is $e = 1$.
Answer for screen readers
The eccentricity of the conic section is $e = 1$.
Steps to Solve
- Rewrite the Equation in Standard Form
Start with the given equation:
$$ 2X^2 + 5Y = 10 $$
Rearrange it to express $Y$ in terms of $X$:
$$ 5Y = 10 - 2X^2 $$
Now, divide everything by 5:
$$ Y = 2 - \frac{2}{5} X^2 $$
This can be rewritten as:
$$ Y = -\frac{2}{5} X^2 + 2 $$
- Identify the Type of Conic Section
The rewritten equation is in the form:
$$ Y = ax^2 + bx + c $$
where $a = -\frac{2}{5}$. Since $a < 0$, this indicates that the conic section is a parabola.
- Calculate the Eccentricity of the Parabola
For parabolas, the eccentricity ($e$) is always:
$$ e = 1 $$
So for our parabola defined by the equation, the eccentricity is:
$$ e = 1 $$
The eccentricity of the conic section is $e = 1$.
More Information
The eccentricity helps to describe the shape of conic sections. For parabolas, the eccentricity is always 1, indicating that they are not closed curves like ellipses or hyperbolas.
Tips
A common mistake is to confuse the type of conic section. Ensure you correctly identify the standard form of the conic before drawing conclusions about its properties.
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