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Chapter 6: Problem 58

Classify the graph of the equation as a circle, a parabola, an ellipse, or a hyperbola. \(x^{2}+y^{2}-4 x-6 y-23=0\)

Short Answer

Expert verified
The given equation represents a Circle with radius \(r = \sqrt{24}\) .

Step by step solution

01

Recognize the general forms of conic sections

A conic section can have one of the four following equations, and each one represents a different figure: \n1. Circle: \(x^2 + y^2 = r^2\), \n2. Ellipse: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \n3. Hyperbola: \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\) or \(-\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), \n4. Parabola: \(y = ax^2+bx+c\) or \(x = ay^2+by+c\).
02

Put the given equation in general forms

Rewriting the given equation \(x^{2}+y^{2}-4 x-6 y-23=0\) into a standard form, we start by grouping the `x` terms and `y` terms together: (x^2 - 4x) + (y^2 - 6y) = 23. Then, we complete the square for `x` and `y` to get \( (x - 2)^2 + (y - 3)^2 = 24 \).
03

Comparing the standard form with conic section forms

Now, we compare the standard form from step 2 with the general forms from step 1. For \((x - 2)^2 + (y - 3)^2 = 24\), upon expansion, we see each of x and y are squared and the coefficients of \(x^2\) and \(y^2\) are equal (both are 1). Therefore, it matches with the equation of a circle, \(x^2 + y^2 = r^2\), with radius \(r = \sqrt{24}\).

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Most popular questions from this chapter

A satellite in a 100-mile-high circular orbit around Earth has a velocity of approximately 17,500 miles per hour. If this velocity is multiplied by \(\sqrt{2}\), the satellite will have the minimum velocity necessary to escape Earth's gravity and it will follow a parabolic path with the center of Earth as the focus (see figure on the next page). (a) Find the escape velocity of the satellite. (b) Find an equation of the parabolic path of the satellite (assume that the radius of Earth is 4000 miles).

Identify the conic as a circle or an ellipse. Then find the center, radius, vertices, foci, and eccentricity of the conic (if applicable), and sketch its graph. \(3 x^{2}+y^{2}+18 x-2 y-8=0\)

Find the eccentricity of the ellipse. \(\frac{x^{2}}{25}+\frac{y^{2}}{36}=1\)

Find the standard form of the equation of the ellipse with the given characteristics. Center: (0,4)\(; a=2 c ;\) vertices: (-4,4),(4,4)

Consider the path of a projectile projected horizontally with a velocity of \(v\) feet per second at a height of \(s\) feet, where the model for the path is \(x^{2}=-\frac{v^{2}}{16}(y-s)\) In this model (in which air resistance is disregarded), \(y\) is the height (in feet) of the projectile and \(x\) is the horizontal distance (in feet) the projectile travels. A cargo plane is flying at an altitude of 30,000 feet and a speed of 540 miles per hour. A supply crate is dropped from the plane. How many feet will the crate travel horizontally before it hits the ground?
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