Problem 1117

Find the center and radius of the circle. Then graph the circle.

x2 + y2 + 3x – 5y – \frac{91}{4} = 0

 

Solution:-

 

To find the center and the radius, complete the square and then write the equation in standard form.

(x – h)2 + (y – k)2 = r

Regroup the terms.

(x2 + 3x ) + (y2 – 5y) = \frac{91}{4}

First complete the square, add \frac{9}{4} to both sides of the equation.

(x2 + 3x + \frac{9}{4}) + (y2 -5y) = \frac{91}{4} + \frac{9}{4}

Now complete square for y2 – 5y.

To complete the square, add \frac{25}{4} to both sides of the equation.

(x + 3x + \frac{9}{4}) + (y – 5y + \frac{25}{4})

= \frac{91}{4}+\frac{9}{4}+\frac{25}{4}

Simplify the right-hand side.

(x + 3x + \frac{9}{4}) + (y – 5y + \frac{25}{4})=\frac{125}{4}

Now write the equation in standard form.

(x – h)2 + (y – k)2 = r2

(x+\frac{3}{2})^{2}+(y-\frac{5}{2})^{2}=\frac{125}{4}

h = –\frac{3}{2}

k = \frac{5}{2}

The radius is \frac{5\sqrt{5}}{2}.

To sum up, the center is (-\frac{3}{2},\frac{5}{2}) and the radius is \frac{5\sqrt{5}}{2}.

To graph the circle, first plot the center (-\frac{3}{2},\frac{5}{2}).

c1

To complete the circle, use the radius to find all the points equidistant from the center.

 

c2

 

 

 

 

 

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