Explanation Given information: The general equation is x 2 + a x + y 2 + b y + c = 0 . Concept used: Equation of circle: The general form of equation of a circle is, ( x − h ) 2 + ( y − k ) 2 = r 2 ....... (1) Calculation: The general equation is given as, x 2 + a x + y 2 + b y + c = 0 ....... (2) Simplify the equation (2) by completing the square as, x 2 + a x + y 2 + b y + c = 0 x 2 + a x + ( a 2 ) 2 − ( a 2 ) 2 + y 2 + b y + ( b 2 ) 2 − ( b 2 ) 2 + c = 0 ( x + a 2 ) 2 + ( y + b 2 ) 2 = ( a 2 ) 2 + ( b 2 ) 2 − c ( x + a 2 ) 2 + ( y + b 2 ) 2 = a 2 4 + b 2 4 − c Further simplify the above equation as, ( x + a 2 ) 2 + ( y + b 2 ) 2 = ( a 2 4 + b 2 4 − c ) 2 Compare the equation (1) with the above obtained equation (3) to obtain the center and radius of the circle as shown below. ( h , k ) = ( − a 2 , − b 2 ) r = a 2 4 + b 2 4 − c For a point, there is no radius therefore the value of radius is zero for a point

7th Edition

ISBN: 9781305115545

Author: James Stewart, Lothar Redlin, Saleem Watson

Publisher: Cengage Learning

See similar textbooks

Question

Chapter 1.2, Problem 113E

To determine

To find:

The simplified form of equation x2+ax+y2+by+c=0 by completing the square and state the condition on the coefficients of the equation such that it represents a point, circle and an empty set. For the case that represents the equation as circle, find radius and center of circle.

Expert Solution & Answer

Chapter 1 Solutions

College Algebra

Ch. 1.1 - The point that is 2 units to the left of the...Ch. 1.1 - If x ¡s positive and y is negative, then the point...Ch. 1.1 - The distance between the points (a, b) and (c, d)...Ch. 1.1 - The point mid ay between (a, b) and (c, d)...Ch. 1.1 - Points in a Coordinate Plane Refer to the...Ch. 1.1 - Points in a Coordinate Plane Refer to the...Ch. 1.1 - Points ¡n a Coordinate Plane Plot the given points...Ch. 1.1 - Points ¡n a Coordinate Plane Plot the given points...Ch. 1.1 - Prob. 9E Ch. 1.1 - Prob. 10E

Knowledge Booster

The simplified form of equation x 2 + a x + y 2 + b y + c = 0 by completing the square and state the condition on the coefficients of the equation such that it represents a point, circle and an empty set. For the case that represents the equation as circle, find radius and center of circle.

Solution Summary: The author explains the simplified form of the equation by completing the square and stating the condition on the coefficients such that it represents a point, circle and an empty set.

To find:

The simplified form of equation x2+ax+y2+by+c=0 by completing the square and state the condition on the coefficients of the equation such that it represents a point, circle and an empty set. For the case that represents the equation as circle, find radius and center of circle.

Chapter 1 Solutions

The simplified form of equation x 2 + a x + y 2 + b y + c = 0 by completing the square and state the condition on the coefficients of the equation such that it represents a point, circle and an empty set. For the case that represents the equation as circle, find radius and center of circle.

Solution Summary: The author explains the simplified form of the equation by completing the square and stating the condition on the coefficients such that it represents a point, circle and an empty set.

To find: The simplified form of equation x2+ax+y2+by+c=0 by completing the square and state the condition on the coefficients of the equation such that it represents a point, circle and an empty set. For the case that represents the equation as circle, find radius and center of circle.

Chapter 1 Solutions

To find:

The simplified form of equation x2+ax+y2+by+c=0 by completing the square and state the condition on the coefficients of the equation such that it represents a point, circle and an empty set. For the case that represents the equation as circle, find radius and center of circle.