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Chapter 7: Problem 11

Identify the conic section represented by the equation. (a) \(2 x^{2}+5 y^{2}=20\) (b) \(x^{2}-y^{2}-8=0\) (c) \(7 y^{2}-2 x=0\) (d) \(x^{2}+y^{2}-25=0\)

Short Answer

Expert verified
(a) Ellipse, (b) Hyperbola, (c) Parabola, (d) Circle.

Step by step solution

01

Equation (a) Analysis

The given equation is \(2x^{2} + 5y^{2} = 20\). This can be rewritten as \(\frac{x^{2}}{10} + \frac{y^{2}}{4} = 1\). This equation is in the standard form of an ellipse, \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a^2 = 10\) and \(b^2 = 4\). Hence, the conic section is an ellipse.
02

Equation (b) Analysis

The given equation is \(x^{2} - y^{2} - 8 = 0\). Rearrange it to \(x^{2} - y^{2} = 8\). This equation is in the form \(x^2 - y^2 = c\), which is the standard form for a hyperbola when not all terms squared are positive or the right side is not zero, confirming it is a hyperbola.
03

Equation (c) Analysis

The given equation is \(7y^{2} - 2x = 0\). Rearrange and solve for \(x\): \(2x = 7y^{2}\) or \(x = \frac{7}{2}y^{2}\). This represents a parabola that opens sideways (horizontal parabola) where \(x\) is expressed in terms of \(y\).
04

Equation (d) Analysis

The given equation is \(x^{2} + y^{2} - 25 = 0\). Rearrange it to \(x^{2} + y^{2} = 25\). This is in the form \(x^2 + y^2 = r^2\), which is the standard form for a circle where \(r^2=25\) indicating that the radius is 5. Hence, it is a circle.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ellipse Equation
An ellipse is one of the basic conic sections, and its standard equation is given by:
  • \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \)
This equation implies two axes: the major axis (along which the distance is larger) and the minor axis. If \(a^2 > b^2\), the ellipse is stretched along the x-axis and vice versa.
Equations of ellipses often appear with different coefficients in front of \(x^2\) and \(y^2\), affecting their shape.
For example, in the equation \(2x^2 + 5y^2 = 20\), when rewritten as \(\frac{x^2}{10} + \frac{y^2}{4} = 1\), it confirms the shape as an ellipse. The ellipse is wider along the x-axis and narrower along the y-axis.
Hyperbola Equation
A hyperbola is another fundamental conic. Its standard form looks like:
  • \( \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \)
This equation represents a pair of symmetrical curves opening either left-to-right or up-and-down.
Unlike an ellipse, a hyperbola has a unique feature called asymptotes, which are lines the hyperbola approaches but never touches.
In equation \(x^2 - y^2 = 8\), it鈥檚 simplified into the hyperbola's form \(\frac{x^2}{8} - \frac{y^2}{8} = 1\). This confirms that the graph is a hyperbola centered at the origin.
Parabola Equation
A parabola is one of the conic sections characterized by its open curve. Typically, its standard form when opening sideways is:
  • \(y^2 = 4ax\) (horizontal opening)
Or when opening up or down:
  • \(x^2 = 4ay\)
Parabolas have a single curve and a vertex, which is the point where they change direction.
In the example \(7y^2 - 2x = 0\), rearranging gives us \(x = \frac{7}{2}y^2\). This represents a sideways-parabola, opening horizontally and using \(y\) to determine its width and orientation.
Circle Equation
A circle is the simplest of these conic sections, described by the equation:
  • \( x^2 + y^2 = r^2 \)
This shows all points that are equidistant from a center point. The radius \(r\) is the distance from the center to any point on the circle.
Given \(x^2 + y^2 - 25 = 0\), rewrite it to the circle form \(x^2 + y^2 = 25\). Here, the radius is 5, identifying the shape as a circle in a simple and straightforward manner.
This form of equation helps quickly recognize the size and position of a circle in geometry.

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Most popular questions from this chapter

Find an orthogonal change of variables that eliminates the cross product terms in the quadratic form \(Q,\) and express \(Q\) in terms of the new variables. $$Q=2 x_{1}^{2}+5 x_{2}^{2}+5 x_{3}^{2}+4 x_{1} x_{2}-4 x_{1} x_{3}-8 x_{2} x_{3}$$

Find a matrix \(P\) that orthogonally diagonalizes \(A,\) and determine \(P^{-1} A P.\) $$A=\left[\begin{array}{llll} 3 & 1 & 0 & 0 \\ 1 & 3 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{array}\right]$$

Prove that each entry on the main diagonal of a skew-Hermitian matrix is either zero or a pure imaginary numbcr.

Let a rectangular \(x^{\prime} y^{\prime} z^{\prime}\) -coordinate system be obtained by rotating a rectangular \(x y z\) -coordinate system counterclockwise about the \(z\) -axis (looking down the \(z\) -axis) through the angle \(\theta=\pi / 4\) (a) Find the \(x^{\prime} y^{\prime} z^{\prime}\) '-coordinates of the point whose \(x y=\) -coordinates are (-1,2,5) (b) Find the \(x y z\) -coordinates of the point whose \(x^{\prime} y^{\prime} z^{\prime}\) -coordinates are (1,6,-3)

(a) A rectangular \(x^{\prime} y^{\prime} z^{\prime}\) 'coordinate system is obtained by rotating an \(x y z\) -coordinate system counterclockwise about the \(y\) -axis through an angle \(\theta\) (looking along the positive \(y\) -axis toward the origin). Find a matrix \(A\) such that $$\left[\begin{array}{l}x^{\prime} \\\y^{\prime} \\\z^{\prime} \end{array}\right]=A\left[\begin{array}{l}x \\\y \\\z \end{array}\right]$$ where \((x, y, z)\) and \(\left(x^{\prime}, y^{\prime}, z^{\prime}\right)\) are the coordinates of the same point in the \(x y z\) - and \(x^{\prime} y^{\prime} z^{\prime}\) 'systems, respectively. (b) Repeat part (a) for a rotation about the \(x\) -axis.
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