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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095

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Calculus

10th Edition
Ron Larson + 1 other
ISBN: 9781285057095
Textbook Problem

Identifying a Conic In Exercises 7-14, identify the conic, analyze the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use a graphing utility to confirm your results.

16 x 2 + 16 y 2 16 x + 24 y 3 = 0

To determine

To graph: The equation 3x2+2y212x+12y+29=0 and also determine the type of conic by analyzing the properties of the equation (center, radius, vertices, foci, eccentricity, directrix, and asymptotes, if possible), and sketch its graph. Use graphing utility to confirm the results.

Explanation

Given:

The equation is 3x2+2y212x+12y+29=0.

Formula used:

The standard equation of an ellipse with center (h,k):

(yk)2a2+(xh)2b2=1

Calculation:

Consider the equation 3x2+2y212x+12y+29=0.

Here,x and y both have squared terms and their coefficients are positive but are not perfect square. Thus, this equation represents an ellipse.

To find the center, write this equation in the form of the standard equation of an ellipse which is given by:

(xh)2a2+(yk)2b2=1

Rearrange the equation as:

3x212x+2y2+12y+29=0

Factoring the coefficients of square terms:

3(x24x)+2(y2+6y)=29x24x+222213+y2+6y+323212=29x24x+2213+y2+6y+3212=29+(4×3)+(9×2)(x2)213+(y+3)212=1

Comparing this with the standard equation of an ellipse with center (h,k):

(yk)2a2+(xh)2b2=1

So, h=2,k=3. Also, a=13,b=12.

So, the center of the ellipse is (2,3).

The larger denominator is b. So, the coordinates of the vertices of an ellipse are (h,k+a) and (h,ka).

So, the vertices are: (2,3+12) and (2,312).

The foci of the ellipse are (h,k+c) and (h,kc) where c=a2b2

So,

c=1213=326=16

Put these values, the foci are obtained as: (2,3+16), (2,316)

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