# A circle has a center that falls on the line #y = 2x +7 # and passes through #(4 ,7 )# and #(1 ,2 )#. What is the equation of the circle?

Equation of the circle is

Additionally, the circle's equation is

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To find the equation of the circle, you need to determine the center (h, k) and the radius (r) using the given points.

First, find the midpoint of the line segment between the two given points to get the center of the circle (h, k). The midpoint formula is:

Midpoint = ((x1 + x2) / 2, (y1 + y2) / 2)

Midpoint = ((4 + 1) / 2, (7 + 2) / 2) = (5/2, 9/2)

So, the center of the circle (h, k) is (5/2, 9/2).

Next, find the distance between one of the given points and the center to determine the radius (r). You can use the distance formula:

Distance = √((x2 - x1)^2 + (y2 - y1)^2)

Using (4, 7) as one of the points:

r = √((5/2 - 4)^2 + (9/2 - 7)^2) = √((5/2 - 4)^2 + (9/2 - 14/2)^2) = √((5/2 - 4)^2 + (-5/2)^2) = √((5/2 - 4)^2 + (25/4)) = √((5/2 - 4)^2 + (25/4)) = √((25/4) + (25/4)) = √(50/4) = √(25/2) = √25 / √2 = 5 / √2

Now, you have the center (h, k) = (5/2, 9/2) and the radius r = 5/√2.

Therefore, the equation of the circle is:

(x - h)^2 + (y - k)^2 = r^2

Substituting the values of h, k, and r:

(x - 5/2)^2 + (y - 9/2)^2 = (5/√2)^2

Simplify if necessary.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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- A circle has a center that falls on the line #y = 1/3x +7 # and passes through # ( 3 ,1 )# and #(7 ,9 )#. What is the equation of the circle?
- A circle has a chord that goes from #pi/3 # to #pi/8 # radians on the circle. If the area of the circle is #25 pi #, what is the length of the chord?
- A triangle has corners at #(3 ,1 )#, #(4 ,9 )#, and #(7 ,4 )#. What is the area of the triangle's circumscribed circle?

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