Find the equation of the circle the end points of whose diameter are the centers of the circles x²+y²+6x-14y=1 and x²+y²-4x+10y=2
Answers
First, recognize that the given equation, x² + y² ‒ 12x + 4y + 6= 0, is the general form for the equation of a circle.
To find the diameter d of the given circle, we’ll have to first convert the given equation, x² + y² ‒ 12x + 4y + 6 = 0, from its current less informative and less useful general form into the more desired standard formfor the equation of a circle by using the method of “completing the square” on its x-terms and on its y-terms as follows:
x² + y² ‒ 12x + 4y + 6 = 0 (given)
Collecting and grouping the x-terms and y-terms together, we get:
(x² ‒ 12x ) + (y² + 4y ) + 6= 0
“Completing the square” in each quadratic grouping, we have:
(x² ‒ 12x + 36) + (y² + 4y + 4) + 6 = 0 + 36 + 4
(x² ‒ 12x + 36) + (y² + 4y + 4) + 6 ‒ 6 = 0 + 36 + 4 ‒ 6
(x² ‒ 12x + 36) + (y² + 4y + 4) = 34
Factoring on the left side, we have:
(x ‒ 6)² + (y + 2)² = 34 is the standard form for the equation of
a circle i.e., (x ‒ h)² + (y ‒ k)² = r², with
center (h, k) and radius r, where, for this
problem, h = 6, k = ‒2, and radius r = √(34).
The radius of a circle is a line segment joining the center of the circle to a point on the circle, and the diameter of a circle is a line segment that passes through the center of the circle and joins two points on the circle; therefore, the length of the diameter is twice the length of the radius for any given circle, and, consequently, the formula for finding the diameter of a circle is: d = 2r; therefore, the diameter of the given circle is found as follows:
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