Comparing - with addition and subtraction

Learning Objectives Distinguish between the standard equations of an ellipse and a hyperbola based on the use of addition versus subtraction. Identify an equation as representing an ellipse because of the addition sign between the squared terms. Convert the general form of an ellipse's equation to its standard form by completing the square. Determine the orientation (horizontal or vertical) of an ellipse from its standard equation. Calculate the center, vertices, co-vertices, and foci of an ellipse given its equation. Articulate how the relationship c^2 = a^2 - b^2 for an ellipse contrasts with the formula for a hyperbola. Why does a plus sign create a closed loop like a planet's orbit, while a minus sign creates the open path of a comet slingshotting around the su...

TermDefinitionExample EllipseA conic section defined as the set of all points (x, y) in a plane, the sum of whose distances from two fixed points (the foci) is constant. Its standard equation involves the sum of squared terms.The equation (x^2 / 25) + (y^2 / 9) = 1 represents an ellipse. Standard Equation of an Ellipse (Centered at Origin)The form (x^2 / a^2) + (y^2 / b^2) = 1 or (x^2 / b^2) + (y^2 / a^2) = 1, where a > b > 0. The key feature is the '+' sign.For (x^2 / 16) + (y^2 / 7) = 1, a^2 = 16 and b^2 = 7. Major and Minor AxesThe major axis is the longer diameter of an ellipse, passing through its foci. The minor axis is the shorter diameter. The lengths are 2a and 2b, respectively.In (x^2 / 25) + (y^2 / 9) = 1, the major axis is horizontal with length 2a = 10, and th...

Standard Equation of a Horizontal Ellipse \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1, \text{ where } a > b Use this form when the ellipse is wider than it is tall. The center is (h, k), the major axis is horizontal, and a^2 is the larger denominator, located under the x-term. Standard Equation of a Vertical Ellipse \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1, \text{ where } a > b Use this form when the ellipse is taller than it is wide. The center is (h, k), the major axis is vertical, and a^2 is the larger denominator, located under the y-term. Focal Distance Formula for an Ellipse c^2 = a^2 - b^2 Use this formula to find the distance 'c' from the center to each focus. Remember, for an ellipse, you subtract the smaller squared denominator from t...