Question

Write each standard equation in general form. $$\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$$

Short answer

The general form of the given equation \(\frac{x^2}{25} + \frac{y^2}{9} = 1\) is \(9x^2 + 25y^2 - 225 = 0\).

Step-by-step solution

Understand the Standard Form of an Ellipse

The given equation is in the standard form of an ellipse which is \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\). Here, \(a\) and \(b\) are the lengths of the semi-major and semi-minor axes, respectively. The task is to convert this standard form into the general form, which is \(Ax^2 + By^2 + Cx + Dy + E = 0\) where \(A\), \(B\), \(C\), \(D\), and \(E\) are constants.

Multiply Both Sides by the Common Denominator

To remove the fractions, multiply both sides of the equation by the least common multiple of the denominators, which is \(25\times9 = 225\). \[225 \left(\frac{x^2}{25} + \frac{y^2}{9}\right) = 225\times1\] which simplifies to \[9x^2 + 25y^2 = 225\].

Rewrite the Equation in General Form

The general form of the equation does not have fractions or a constant on the right side. Our current equation \(9x^2 + 25y^2 = 225\) is almost in general form but we need to set the right side to zero. Thus, subtract \(225\) from both sides to obtain the general form: \[9x^2 + 25y^2 - 225 = 0\].

Key concepts

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. Depending on the angle of the plane to the cone's axis, the intersection can result in different shapes: a circle, an ellipse, a parabola, or a hyperbola. An ellipse, one of these shapes, is formed when the plane's angle is such that it cuts through both naps of the cone but does so at an angle to the axis that is not perpendicular. This means an ellipse is a conic section that is not as open as a parabola and not as closed as a circle. It can be seen in various applications such as planetary orbits and architectural designs.

When discussing conic sections, understanding the basic terminology is crucial. The major axis is the longest diameter of the ellipse, and the minor axis is the shortest. The points where the ellipse is widest horizontally are called the vertices, and the points where it is narrowest vertically are the co-vertices. Additionally, each ellipse has two foci, which are points inside the ellipse such that the sum of the distances to the foci from any point on the ellipse is constant. This property is what makes ellipses unique and interesting to study.

Ellipse Equations

Ellipse equations are mathematical representations of the shape and position of an ellipse. They typically fall into two categories: standard form and general form. The standard form of an ellipse's equation is very informative because it shows the radii of the ellipse directly. In the example given above, \frac{x^2}{25} + \frac{y^2}{9} = 1\, you can easily see that the length of the semi-major axis, \(a\), is 5 units, and the length of the semi-minor axis, \(b\), is 3 units. These values are derived from the denominators of the fractions under \(x^2\) and \(y^2\), respectively, when in the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).

To decipher an ellipse's properties, different parameters are used: \(a\) and \(b\) denote the lengths from the center to the ellipse's vertices and co-vertices, respectively. The center of the ellipse (\(h,k\)) can also be varied in the standard form, but in the version given, it is at the origin (0,0). Recognizing these elements within the equation of an ellipse is key to understanding and graphing the shape effectively.

General Form of an Ellipse

The general form of an ellipse takes a more encompassing approach and is written as \(Ax^2 + By^2 + Cx + Dy + E = 0\), where the variables represent constants that dictate the shape and position of the ellipse. The transition from standard form to general form, as demonstrated in the exercise, is a process of generalizing an ellipse's equation so that it becomes compatible with different methods and applications, such as integrating with other equations or analyzing the conic section's orientation and location in a more complex coordinate system.

The general form of an ellipse is necessary for certain types of analysis and calculations. For example, if you want to determine the eccentricity, find the foci's locations, or transform the ellipse through various algebraic manipulations, the general form is preferable. In the case of the example provided, simplifying the equation by multiplying through by the least common multiple and setting the equation equal to zero transforms it into a format much more amenable to further algebraic operation.