# A circle has a center that falls on the line #y = 11/7x +8 # and passes through # ( 9 ,1 )# and #(8 ,4 )#. What is the equation of the circle?

locating the circle's center:

We are aware that the distances between the center and each of the two points on the circle will be equal because we know two of the points.

Set each of these points equal by using the formula for the distance between the center and each of these points:

Simplify inside the square roots and square both sides.

Expand.

This provides us with

The next thing we must do is figure out what to do next. In the end, we want to figure out the circle's equation, and the standard form of that equation is

Knowing the center of the circle allows us to know that

Finding the radius of the circle:

Formulating the circle's equation:

Applying the formula

We have

Slightly streamlined:

Visual inspection:

y-11/7x-8)((x-9)^2+(y-1)^2-.2)((x-8)^2+(y-4)^2-.2)((x+175/26)^2+(y+67/26)^2-87965/338)(x+175/26)^2+(y+67/26)^2=0 [-41.56, 31.5, -21.63, 14.92]}

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To find the equation of the circle, we first need to find the center of the circle. Since the center falls on the line (y = \frac{11}{7}x + 8), we can equate the equation of the line with the general form of a circle's equation, which is ((x - h)^2 + (y - k)^2 = r^2), where ((h, k)) is the center of the circle and (r) is the radius.

Given that the circle passes through the points ((9, 1)) and ((8, 4)), we can find the midpoint of these two points, which will be the center of the circle.

Midpoint formula: [h = \frac{x_1 + x_2}{2}] [k = \frac{y_1 + y_2}{2}]

Substitute the coordinates of the given points: [h = \frac{9 + 8}{2} = \frac{17}{2}] [k = \frac{1 + 4}{2} = \frac{5}{2}]

So, the center of the circle is ((\frac{17}{2}, \frac{5}{2})).

Now, we need to find the radius of the circle. We can use the distance formula between the center of the circle and one of the given points. Let's use the point ((9, 1)) for this calculation.

Distance formula: [r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}]

Substitute the coordinates: [r = \sqrt{(9 - \frac{17}{2})^2 + (1 - \frac{5}{2})^2}] [r = \sqrt{(\frac{1}{2})^2 + (-\frac{3}{2})^2}] [r = \sqrt{\frac{1}{4} + \frac{9}{4}}] [r = \sqrt{\frac{10}{4}} = \frac{\sqrt{10}}{2}]

Now, we have the center ((\frac{17}{2}, \frac{5}{2})) and the radius (\frac{\sqrt{10}}{2}). We can plug these values into the general form of the circle's equation to obtain the final equation:

[(x - \frac{17}{2})^2 + (y - \frac{5}{2})^2 = (\frac{\sqrt{10}}{2})^2]

This simplifies to:

[(x - \frac{17}{2})^2 + (y - \frac{5}{2})^2 = \frac{10}{4}]

Therefore, the equation of the circle is:

[(x - \frac{17}{2})^2 + (y - \frac{5}{2})^2 = \frac{5}{2}]

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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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