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Chapter 10: Problem 7

What is the equation of the standard ellipse with vertices at \((\pm a, 0)\) and foci at \((\pm c, 0) ?\)

Short Answer

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Question: Find the equation of a standard ellipse with vertices at (卤a, 0) and foci at (卤c, 0). Answer: The equation of a standard ellipse with vertices at (卤a, 0) and foci at (卤c, 0) is given by: \(\frac{x^2}{a^2} + \frac{y^2}{a^2-c^2} = 1\)

Step by step solution

01

Recall the general equation of a standard ellipse

The equation of a standard ellipse with center at the origin and major axis aligned along the x-axis is given by: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) Here, a is the distance from the center of the ellipse to a vertex along the major axis, and b is the distance from the center of the ellipse to a co-vertex along the minor axis. Since the vertices are given as \((\pm a, 0)\), we know the major axis is aligned along the x-axis, and the center of the ellipse will be at the origin (0,0).
02

Recall the relationship between the foci, vertices, and co-vertices

For a standard ellipse, the distance between the foci and the center is related to a and b by the equation: \(c^2 = a^2 - b^2\) In this problem, we are given that the foci are at the points \((\pm c,0)\).
03

Find the value of b

Since we are given values for a and c, we can use the relationship between the foci, vertices, and co-vertices to find the value of b. From step 2, we know that: \(c^2 = a^2 - b^2\) Rearrange the equation to solve for b^2: \(b^2 = a^2 - c^2\)
04

Substitute the values of a and c into the general equation

Now that we have an expression for b^2 in terms of a and c, we can substitute these values into the general ellipse equation obtained in Step 1: \(\frac{x^2}{a^2} + \frac{y^2}{a^2-c^2} = 1\) This is the equation of the standard ellipse with vertices at \((\pm a, 0)\) and foci at \((\pm c, 0)\).

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Most popular questions from this chapter

Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The lower half of the circle centered at \((-2,2)\) with radius 6 oriented in the counterclockwise direction

Consider the parametric equations $$ x=a \cos t+b \sin t, \quad y=c \cos t+d \sin t $$ where \(a, b, c,\) and \(d\) are real numbers. a. Show that (apart from a set of special cases) the equations describe an ellipse of the form \(A x^{2}+B x y+C y^{2}=K,\) where \(A, B, C,\) and \(K\) are constants. b. Show that (apart from a set of special cases), the equations describe an ellipse with its axes aligned with the \(x\) - and \(y\) -axes provided \(a b+c d=0\) c. Show that the equations describe a circle provided \(a b+c d=0\) and \(c^{2}+d^{2}=a^{2}+b^{2} \neq 0\)

An idealized model of the path of a moon (relative to the Sun) moving with constant speed in a circular orbit around a planet, where the planet in turn revolves around the Sun, is given by the parametric equations $$x(\theta)=a \cos \theta+\cos n \theta, y(\theta)=a \sin \theta+\sin n \theta.$$ The distance from the moon to the planet is taken to be \(1,\) the distance from the planet to the Sun is \(a,\) and \(n\) is the number of times the moon orbits the planet for every 1 revolution of the planet around the Sun. Plot the graph of the path of a moon for the given constants; then conjecture which values of \(n\) produce loops for a fixed value of \(a\) a. \(a=4, n=3\) b. \(a=4, n=4 \) c. \(a=4, n=5\)

A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and \(3,\) respectively, and that Earth completes one orbit in one year while Mars takes two years. When \(t=0,\) Earth is at (2,0) and Mars is at (3,0) both orbit the Sun (at (0,0) ) in the counterclockwise direction. The position of Mars relative to Earth is given by the parametric equations \(x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t\) a. Graph the parametric equations, for \(0 \leq t \leq 2\) b. Letting \(r=(3-4 \cos \pi t),\) explain why the path of Mars relative to Earth is a lima莽on (Exercise 89).

Show that an ellipse and a hyperbola that have the same two foci intersect at right angles.
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