For Exercise 41-54, write the equation in the form x − h 2 + y − k 2 = c . Then if the equation represents a circle, identify the center and radius. If the equation represents a degenerate case, give the solution set. (See Examples 3-4) x 2 + y 2 + 12 x − 14 y + 84 = 0
For Exercise 41-54, write the equation in the form x − h 2 + y − k 2 = c . Then if the equation represents a circle, identify the center and radius. If the equation represents a degenerate case, give the solution set. (See Examples 3-4) x 2 + y 2 + 12 x − 14 y + 84 = 0
Solution Summary: The author explains how the equation, x2+y2,+12x-14y+84=0, represents a circle with the center and the radius of 1.
For Exercise 41-54, write the equation in the form
x
−
h
2
+
y
−
k
2
=
c
.
Then if the equation represents a circle, identify the center and radius. If the equation represents a degenerate case, give the solution set. (See Examples 3-4)
Recall that an equation of a circle can be written in standard form (x-h)² + (y-k)² = r, where (h, k) is the
center and r is the radius.
After expanding both squares, moving all terms to the left-hand side, and combining like terms, the equation
can also be written in the form x² + y² + Ax+By+ C = 0, where A, B, and C are constants.
a. Find an equation of the form x2 + y² + Ax+By+C =0 for the circle that passes through the points
(6, 0), (2, 2), and (7,-3).
To do so, find the values of A, B, and C by writing and solving a system of 3 linear equations.
System:
Show the steps of solving the system.
Equation in Standard Form:
Solution:
b. Rewrite the equation found in part (a) in standard form using the technique of completing the square.
c. Use your result in part (b) to determine the center and radius of the circle.
Center:
A =
B =
C =
Radius:
In three-space, there are four possibilities for the intersection of two lines.If one line is Z-axis, write a possible equation for the second line in each of the four cases.
Can you explain why you used the certain equations?
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