Find the standard form of the equation of each ellipse satisfying the given conditions. Foci: (-

step 1 We have Ptollar number 25 and this we have to find the standard form of the equation of the elli

Find the standard form of the equation of each ellipse...

Key Concepts

Ellipse

An ellipse is a set of points in a plane such that the sum of the distances from any point on the ellipse to two fixed points (known as the foci) is constant. This geometric definition leads to unique properties and a smooth, closed curve, making ellipses a fundamental conic section in mathematics.

Standard Equation of an Ellipse

The standard form of an ellipse's equation characterizes its location and shape. Typically, the equation is written as (x - h)²/a² + (y - k)²/b² = 1 for an ellipse centered at (h, k) with a horizontal major axis, or with the denominators swapped for a vertical major axis. In these equations, a and b are the lengths of the semi-major and semi-minor axes, respectively.

Foci

The foci of an ellipse are two special points located along the major axis. For any point on the ellipse, the sum of the distances to the two foci is constant. The positions of the foci relative to the center define the 'eccentricity' of the ellipse, which measures how elongated the ellipse is.

Vertices

Vertices are the intersection points of the ellipse with its major axis. They represent the furthest points from the ellipse's center along the direction of the major axis. Knowing the vertices helps determine the length of the semi-major axis, which is central to forming the standard equation of the ellipse.

Relationship between a, b, and c

In any ellipse, the distance from the center to a focus is denoted by c, and the lengths of the semi-major and semi-minor axes are denoted by a and b, respectively. These parameters are related by the equation c² = a² - b² (for an ellipse with a horizontal major axis). This relationship is crucial for calculating the unknown parameter when the positions of the foci and vertices are given.

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