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Chapter 10: Problem 16

Find the standard form of the equation of the ellipse with the given characteristics and center at the origin. Vertices: (0,卤8)\(;\) foci: (0,卤4)

Short Answer

Expert verified
The standard form of the equation of the ellipse is \[ \frac{x^2}{16} + \frac{y^2}{64} = 1 \] .

Step by step solution

01

Determine the lengths of the semi-major and semi-minor axes

The vertices of the ellipse are given as (0,卤8), so the semi-major axis length is 8. The foci of the ellipse are given as (0,卤4), so the distance between the center and the foci, which is the semi-minor axis, is 4. Therefore, \( a = 8 \) and \( b = 4 \).
02

Identify the orientation

Since the vertices and foci are aligned along the y-axis, the ellipse is vertically oriented. This means that \( a \), the length of the semi-major axis, is under the y term in the equation, and \( b \), the length of the semi-minor axis, is under the x term.
03

Write the equation of the ellipse

Substitute the values of a and b into the standard equation of the ellipse. Therefore, the equation of the ellipse is \[ \frac{x^2}{4^2} + \frac{y^2}{8^2} = 1 \] , which simplifies to \[ \frac{x^2}{16} + \frac{y^2}{64} = 1 \] .

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Most popular questions from this chapter

Convert the rectangular equation to polar form. Assume \(a<0\) $$y=-\sqrt{3} x$$

Find a polar equation of the conic with its focus at the pole. $$\begin{array}{cc} \text{Conic} & \text{Vertex or Vertices} \\\ \text{Parabola} &\left(10, \frac{\pi}{2}\right)\end{array}$$

Determine whether the statement is true or false. Justify your answer. If \(\left(r_{1}, \theta_{1}\right)\) and \(\left(r_{2}, \theta_{2}\right)\) represent the same point in the polar coordinate system, then \(\left|r_{1}\right|=\left|r_{2}\right|\)

Describe the graph of the polar equation and find the corresponding rectangular equation. Sketch its graph. $$r=6$$

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