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Chapter 10: Problem 15

Use the discriminant to determine whether the graph of the equation is an ellipse (or a circle), a hyperbola, or a parabola. $$2 x^{2}-8 x y+7 y^{2}+x-2 y+1=0$$

Short Answer

Expert verified
The equation represents a hyperbola.

Step by step solution

01

Identify coefficients

Write the given equation in the standard quadratic form and identify the coefficients. The given equation is 2x^2-8xy+7y^2+x-2y+1=0 Here, we identify the coefficients as follows: A=2, B=-8, C=7.
02

Calculate the Discriminant

Use the formula for the discriminant of a conic section: D = B^2 - 4AC Plug in the values of A, B, and C: D = (-8)^2 - 4(2)(7) Calculate this expression to find the discriminant.
03

Simplify the Discriminant

Continue simplifying the discriminant: D = 64 - 56 = 8 So, the discriminant D is 8.
04

Determine the Type of Conic Section

Use the value of the discriminant to determine the type of conic section: - If D < 0, the conic is an ellipse (or a circle).- If D = 0, the conic is a parabola.- If D > 0, the conic is a hyperbola. Since D = 8 > 0, the conic section is a hyperbola.

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Key Concepts

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

Ellipse
An ellipse is a conic section that forms a closed curve, resembling an elongated circle. In terms of equations, it is defined as follows:
When the equation of the conic section has its discriminant (\text{D}) less than 0 (\text{D} < 0), it represents an ellipse or a circle.
The standard equation of an ellipse is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] Here, \text{a} and \text{b} are the lengths of the semi-major and semi-minor axes, respectively. Key features of an ellipse include:
  • Major Axis: The longest diameter of the ellipse, running through its center and both foci.
  • Minor Axis: The shortest diameter, perpendicular to the major axis at the center.
  • Foci (plural of focus): Two fixed points inside the ellipse such that the sum of the distances from any point on the ellipse to the foci is constant.
Understanding the geometrical properties is crucial as these affect the shape and size of the ellipse.
Hyperbola
A hyperbola is another type of conic section which is defined by its open, curve-like appearance. Unlike an ellipse, a hyperbola consists of two separate curves.
This is determined when the discriminant (\text{D}) of the conic section is greater than 0 (\text{D} > 0). The standard equation of a hyperbola is: \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] or its rotated counterpart. Key characteristics of a hyperbola include:
  • Transverse Axis: This is the line segment that passes through the foci of the hyperbola.
  • Conjugate Axis: Perpendicular to the transverse axis, it does not intersect the hyperbola.
  • Foci: These are two fixed points that are used in the definition of the hyperbola. The difference between the distances from any point on a hyperbola to the foci is a constant.
  • Vertices: Found on the transverse axis, these are the points where each branch of the hyperbola intersects the axis.
Hyperbolas are often identified by their 'hourglass' shape and are useful in fields such as physics and astronomy.
Parabola
A parabola is a unique type of conic section characterized by its open, U-shaped curve. This occurs when the discriminant (\text{D}) of the equation equals zero (\text{D} = 0). The standard equation of a parabola, when aligned with its vertex at the origin, is given by: \[ y = ax^2 \] Here, \text{a} determines the width and direction of the parabola. Key features include:
  • Vertex: The most significant point, representing the lowest or highest point on the parabola, depending on its orientation.
  • Focus: A fixed point inside the parabola. Every point on a parabola is equidistant from the focus and the directrix.
  • Directrix: A line outside the parabola. Used with the focus, it helps define the curve.
  • Axis of Symmetry: A vertical line through the vertex that divides the parabola into two mirrored halves.
Parabolas have important applications in fields like physics, engineering, and astronomy, especially in the behavior of projectile motion and reflective properties.

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Most popular questions from this chapter

Solve. [ 5.5] $$5^{x}=27$$

a) Determine whether it is one-to-one. [5.1] b) If it is one-to-one, find a formula for the inverse. [ 5.1] $$f(x)=\sqrt{x+4}$$

Find an equation of the circle that passes through the points \((2,4)\) and \((3,3)\) and whose center is on the line \(3 x-y=3\)

Find an equation of an ellipse centered at the origin that passes through the points \((1, \sqrt{3 / 2})\) and \((\sqrt{3}, 1 / 2)\)

Solve using a graphing calculator. Find all real solutions. $$\begin{aligned} &y-e^{-x}=1\\\ &y=2 x+5 \end{aligned}$$
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