What conclusion can you make about the coordinates of the vertex and the length of the radius of a circle?
(x – 2)2 + (y – 3)2.
The numerical side, the 16, is the square of the radius, so it actually indicates 16 = r2 = 42, so the radius is r = 4. Reading from the squared-variable parts, the center is at (h, k) = (2, 3).
State the radius and center of the circle with equation 25 = x2 + (y + 3)2.
The numerical side tells me that r2 = 25, so r =
5. The x-squared part is really (x – 0)2, so h = 0. The temptation is to read off the "3" from the y-squared part and conclude that k is 3, but this is wrong. The center-vertex form has subtraction in it, so I need to convert first to that form.
y + 3 = y – (–3)
So the y-coordinate of the center is actually k = –3.
radius r = 5, center (h, k) = (0, –3)
Warning: It is very easy to forget that sign in the middle of the squared parts. Don't be careless!
In the previous examples, information was extracted from a given equation. You'll also need to be able to work from given information backwards to find an equation.
Find an equation for the circle with center (h, k) = (4, –2) and radius r = 10.
I'll just plug the center and radius into the center-radius form:
(x – (4))2 + (y – (–2))2 = 102
(x – 4)2 + (y + 2)2 = 100
Since no particular form of the equation was specified, the above is an acceptable answer. If your book specifies some other format, then you may need to multiply things out:
x2 – 8x + 16 + y2 + 4y + 4 = 100
x2 + y2 – 8x + 4y + 20 = 100
x2 + y2 – 8x + 4y – 80 = 0
Keep in mind that there is no standard meaning to the term "standard form". If your book specifies some other form, memorize that form for your tests.
Answers & Comments
What conclusion can you make about the coordinates of the vertex and the length of the radius of a circle?
(x – 2)2 + (y – 3)2.
y + 3 = y – (–3)
radius r = 5, center (h, k) = (0, –3)
I'll just plug the center and radius into the center-radius form:
(x – (4))2 + (y – (–2))2 = 102
(x – 4)2 + (y + 2)2 = 100
x2 – 8x + 16 + y2 + 4y + 4 = 100
x2 + y2 – 8x + 4y + 20 = 100
x2 + y2 – 8x + 4y – 80 = 0
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