Find the eccentricity for the equation 2x^2 + 5 - 10 = 0.
Understand the Problem
The question is asking how to find the eccentricity of the conic section represented by the equation 2x^2 + 5 - 10 = 0. This involves identifying the type of conic section (e.g., ellipse, hyperbola, etc.) and using the relevant formula for eccentricity based on the equation's standard form.
Answer
$ e = 1 $
Answer for screen readers
The eccentricity of the conic section is ( e = 1 ).
Steps to Solve
- Rearranging the equation
First, we'll simplify the given equation (2x^2 + 5 - 10 = 0).
This simplifies to: $$ 2x^2 - 5 = 0 $$
- Isolate the quadratic term
Next, isolate the (x^2) term by adding 5 to both sides of the equation.
This gives us: $$ 2x^2 = 5 $$
- Divide by the coefficient of x²
Now, divide both sides by 2 to solve for (x^2).
We obtain: $$ x^2 = \frac{5}{2} $$
- Identify the conic section type
The equation is now in the form ( x^2 = \frac{5}{2} ). Since it's a simple quadratic equation and represents a parabola opening sideways, we conclude that this is a parabola.
- Find the eccentricity of the parabola
The eccentricity (e) of a parabola is always (1).
Therefore: $$ e = 1 $$
The eccentricity of the conic section is ( e = 1 ).
More Information
In conic sections, eccentricity is a measure of how much the conic deviates from being circular. Parabolas have a unique property where the distance from any point on the parabola to the focus is equal to the distance to the directrix, which results in an eccentricity of exactly (1).
Tips
- Confusing the equation of a conic with other forms; make sure to correctly identify the type (ellipse, hyperbola, or parabola).
- Miscalculating the eccentricity; remember that for a parabola, it's always (1).
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