53–57. Conic sections
a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.
x²/4 + y²/25 = 1

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Identify the general form of the conic section equation. The given equation is \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \). This resembles the standard form of conic sections centered at the origin.

Recall the standard forms: - Ellipse: \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) - Hyperbola: \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \) or \( -\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) - Parabola: Usually involves only one squared term, e.g., \( y^2 = 4ax \) or \( x^2 = 4ay \).

Compare the given equation to these forms. Since both \( x^2 \) and \( y^2 \) terms are positive and added together, and the equation equals 1, it matches the form of an ellipse.

Note the denominators: \(4\) and \(25\) represent \(a^2\) and \(b^2\) respectively, which are positive constants, confirming the ellipse shape.

Conclude that the equation \( \frac{x^2}{4} + \frac{y^2}{25} = 1 \) describes an ellipse.

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Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. The main types are parabolas, ellipses, and hyperbolas, each defined by specific algebraic equations and geometric properties.

Standard Form of Conic Equations

Each conic section has a standard equation form: ellipses and hyperbolas involve sums or differences of squared terms set equal to 1, while parabolas have a single squared term. Recognizing these forms helps identify the conic type.

Ellipse Identification Criteria

An ellipse equation typically has the form (x²/a²) + (y²/b²) = 1 with both denominators positive and the sum of squared terms. This distinguishes it from hyperbolas, which have a subtraction, and parabolas, which have only one squared term.

53–57. Conic sections a. Determine whether the following equations describe a parabola, an ellipse, or a hyperbola.x²/4 + y²/25 = 1