Problem 25 - Algebra 2

Question

The rounded top of the window is the top half of an ellipse. Write an equation for the ellipse if the origin is at the midpoint of the bottom edge of the window. \(16 x^{2}+25 y^{2}+32 x-150 y=159\)

Step-by-Step Solution

Answer

The ellipse equation is \(\frac{(x+1)^2}{25} + \frac{(y-3)^2}{16} = 1\).

Step 1: Rearrange the Equation to Standard Form

We begin with the given equation of the ellipse: \(16x^2 + 25y^2 + 32x - 150y = 159\). First, move all terms to the left-hand side: \(16x^2 + 32x + 25y^2 - 150y - 159 = 0\).

Step 2: Complete the Square for X-Terms

Factor out the coefficient of \(x^2\) from the \(x\) terms: \(16(x^2 + 2x)\). To complete the square, take half of 2 (which is 1), square it to get 1, and add and subtract 16 in the equation: \(16((x+1)^2 - 1) = 16(x+1)^2 - 16\).

Step 3: Complete the Square for Y-Terms

Factor out the coefficient of \(y^2\) from the \(y\) terms: \(25(y^2 - 6y)\). To complete the square, take half of -6 (which is -3), square it to get 9, and add and subtract 225 in the equation: \(25((y-3)^2 - 9) = 25(y-3)^2 - 225\).

Step 4: Adjust and Simplify the Equation

Substitute the completed square terms back into the equation: \[16(x+1)^2 - 16 + 25(y-3)^2 - 225 = 159\].Simplify the constants: \[16(x+1)^2 + 25(y-3)^2 = 400\].

Step 5: Normalize to the Standard Ellipse Equation

Divide every term by 400 to get the equation in standard form:\[\frac{(x+1)^2}{25} + \frac{(y-3)^2}{16} = 1\].This shows the ellipse centered at \((-1, 3)\) with a horizontal semi-axis of 5 and a vertical semi-axis of 4.

Key Concepts

Completing the Square

Completing the square is a method used to rewrite quadratic equations. This technique is crucial in finding the standard form of conic sections such as ellipses. In the provided exercise, you have an equation of the form:\[16x^2 + 32x + 25y^2 - 150y = 159.\]To complete the square, focus on terms involving each variable separately. For the \(x\)-terms \(16x^2 + 32x\), factor out 16, giving:

• \(16(x^2 + 2x)\)

Then, take half of the coefficient of \(x\) (which is 2), square it (getting 1), and incorporate it to complete the square:\[16((x+1)^2 - 1) = 16(x+1)^2 - 16.\]Similarly, for the \(y\)-terms, factor out 25:

• \(25(y^2 - 6y)\)

Take half of the coefficient of \(y\) (which is -6), square it (getting 9), and incorporate it:\[25((y-3)^2 - 9) = 25(y-3)^2 - 225.\]With these steps, you've transformed the quadratic parts of the original equation, setting the stage for converting the form into a standard ellipse equation.

Standard Form of an Ellipse

The standard form of an ellipse's equation is crucial for understanding its properties, such as its center and the lengths of its axes. After completing the square, your transformed equation becomes:\[16(x+1)^2 + 25(y-3)^2 = 400.\]To rewrite this in standard form, divide each term by 400, resulting in:\[\frac{(x+1)^2}{25} + \frac{(y-3)^2}{16} = 1.\]This form reveals essential information about the ellipse:

• The center is at the point \((-1, 3)\).
• The semi-major axis is along the horizontal with a length of 5 units (\(\sqrt{25}\)).
• The semi-minor axis is vertical with a length of 4 units (\(\sqrt{16}\)).

The standard form is beneficial as it helps us quickly identify such features of the ellipse, enhancing our ability to graph and analyze it.

Coordinate Geometry

Coordinate geometry involves graphing equations and interpreting geometric properties through algebraic expressions. It's a bridge between algebra and geometry. For ellipses, coordinate geometry helps us visualize their shape and location on a Cartesian plane.In cases like our exercise, once the equation has been simplified to the standard form, coordinate geometry principles can be applied:

• The equation \(\frac{(x+1)^2}{25} + \frac{(y-3)^2}{16} = 1\) indicates an ellipse centered at \((-1, 3)\).
• Plot this point as the center of the ellipse.
• Use the lengths of the axes derived from the standard form to sketch the ellipse: horizontal axis 10 units long (5 units each side from the center), vertical axis 8 units long (4 each side).

By interpreting the equation through these steps, coordinate geometry allows us to transform an algebraic equation into a visual representation, essential for a deeper understanding of geometric shapes in algebra.

Analytical Geometry

Analytical geometry, also known as coordinate geometry, underpins many mathematical concepts by combining algebra and geometry. This field facilitates precise calculations and visualizations for geometric figures, like circles and ellipses.Imagine working on the given ellipse equation:\[16x^2 + 25y^2 + 32x - 150y = 159.\]Through tools seen in analytical geometry, we convert complex polynomial equations into the understandable standard form as shown:\[\frac{(x+1)^2}{25} + \frac{(y-3)^2}{16} = 1.\]Analytical geometry provides the structure needed to deduce characteristics such as:

• The central point of the ellipse \((-1, 3)\).
• Interrelationships in dimensions—the semi-major and minor axis lengths.
• How changes to the equation affect its geometrical representation.

This branch of study makes it easier to not only solve such problems but also to understand the spatial relationships and graphical representations they represent.