   Chapter 12.CT, Problem 16CT ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742

#### Solutions

Chapter
Section ### Algebra and Trigonometry (MindTap ...

4th Edition
James Stewart + 2 others
ISBN: 9781305071742
Textbook Problem

# (a)Use the discriminant to determine whether the graph of the following equation is a parabola, an ellipse, or a hyperbola: 5 x 2 + 4 x y + 2 y 2 = 18 (b) Use rotation of axes to eliminate the x y -term in the equation.(c) Sketch a graph of the equation.(d) Find the coordinates of the vertices of this conic (in the x y -coordinate system).

To determine

(a).

To find:

Whether the graph of the following equation is a parabola, an ellipse, or a hyperbola:

5x2+4xy+2y2=18

Explanation

Given:

An equation of a conic section as, 5x2+4xy+2y2=18.

Approach:

Compare the given equation of the conic with the general equation of the conic,

Ax2+Bxy+Cy2+Dx+Ey+F=0

The quantity B24AC is known as discriminant of the above equation,

i. If B24AC>0, then the above equation is of hyperbola equation.

ii. If B24AC=0, then the above equation is of parabola equation.

iii. If B24AC<0, then the above equation is of an ellipse equation.

Calculation:

Given equation of conic is 5x2+4xy+2y2=18

To determine

(b).

To find:

The equation of the given conic after removing the xy-term, use the rotation of axes.

To determine

(c).

To Sketch:

A graph of the equation 5x2+4xy+2y2=18.

To determine

(d).

To find:

The coordinates of the vertices of the conic 5x2+4xy+2y2=18.

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