a) for the equation (x-1)^2 / 25+y^2 / 9=1, find the y -coordinates when x=-4,-3,-2,-1,0,1,2,3,4

step 1 Giving the equation x squared over 25 plus y squared over 9 equals 1, since we're adding, it's a

A) for the equation (x-1)^2 / 25+y^2 / 9=1, find the y...

a) for the equation $(\mathrm{x}-1)^{2} / 25+\mathrm{y}^{2} / 9=1$, find the $\mathrm{y}$ -coordinates when $\mathrm{x}=-4,-3,-2,-1,0,1,2,3,4,5$ and 6
b) plot the points $(x, y)$ obtained in (a), and sketch the results.

Key Concepts

Symmetry in Conic Sections

Ellipses exhibit symmetry about both their horizontal and vertical axes. This property means that for every point (x, y) on the ellipse, there exists a corresponding point (x, -y) (assuming the ellipse is centered on the x-axis) or their respective symmetric counterparts when centered elsewhere. Recognizing symmetry simplifies the computation and sketching process by reducing the effort needed to evaluate every point explicitly.

Substitution and Solving for Coordinates

When specific x-values are provided, the corresponding y-coordinates can be determined by substituting these values into the ellipse equation. This process involves isolating the y-term, often leading to two solutions for y for each x (one positive and one negative), which represent points lying symmetrically above and below the center of the ellipse.

Graphing and Sketching Techniques

Graphing an ellipse involves plotting a set of calculated points and then sketching the smooth, continuous curve that passes through them. This process not only aids in visualizing the geometric shape but also verifies the accuracy of computed coordinates. The ability to convert algebraic equations into graphical representations is a fundamental skill in coordinate geometry.

Ellipse

An ellipse is a type of conic section defined as the set of points for which the sum of the distances from two fixed focal points is constant. The standard form of an ellipse centered at (h, k) is (x - h)²/a² + (y - k)²/b² = 1, where a and b represent the semi-major and semi-minor axes respectively. Understanding ellipses is essential as they appear in various fields, from orbital mechanics to optics.

Standard Form of an Ellipse Equation

The standard form of an ellipse helps to readily identify its center, axes lengths, and orientation. For an ellipse given by (x - h)²/a² + (y - k)²/b² = 1, the values of h and k shift the center from the origin, while a and b determine the horizontal and vertical stretches of the ellipse. Recognizing this form is crucial for graphing and analyzing the ellipse's geometric properties.

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