# A circle has a center that falls on the line #y = 2/7x +3 # and passes through # ( 1 ,4 )# and #(6 ,1 )#. What is the equation of the circle?

#(x-133/29)^2+(y-125/29)^2 = 10897/29^2#

Or if you prefer:

#(29x-133)^2+(29y-125)^2 = 10897#

So:

Thus, the following is how our circle's equation can be expressed:

Thus, the circle's equation can be expressed as follows:

graph{((x-133/29)^2+(y-125/29)^2-10897/29^2)((x-1)^2+(y-4)^2-0.02)((x-6)^2+(y-1)^2-0.02)(y - 2/7x-3)((x-133/29)^2+(y-125/29)^2-0.04)=0 [-6.58, 13.42, -1, 9]}

By signing up, you agree to our Terms of Service and Privacy Policy

The equation of the circle can be found by using the formula for the general form of the equation of a circle, 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 center of the circle falls on the line (y = \frac{2}{7}x + 3), we need to find the coordinates of the center ((h, k)).

First, we find the midpoint of the line segment connecting the given points ((1, 4)) and ((6, 1)). The midpoint formula is ((\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2})).

Midpoint coordinates: [ \left(\frac{1 + 6}{2}, \frac{4 + 1}{2}\right) = (3.5, 2.5) ]

Now, since the center of the circle lies on the line (y = \frac{2}{7}x + 3), we set (y) equal to (2.5) and solve for (x): [ 2.5 = \frac{2}{7}x + 3 ] [ x = \frac{7}{2}(2.5 - 3) = -\frac{7}{2} \times \frac{1}{2} = -\frac{7}{4} ]

Now, substituting (x = -\frac{7}{4}) into the equation of the line gives us the (y)-coordinate: [ y = \frac{2}{7} \times (-\frac{7}{4}) + 3 = -1 + 3 = 2 ]

Therefore, the center of the circle is ((- \frac{7}{4}, 2)).

Next, we need to find the radius of the circle, which is the distance from the center to any point on the circle. We can use the distance formula: [ r = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} ]

Using the points ((1, 4)) and ((-\frac{7}{4}, 2)) for the radius calculation: [ r = \sqrt{(1 + \frac{7}{4})^2 + (4 - 2)^2} = \sqrt{(\frac{11}{4})^2 + 2^2} = \sqrt{\frac{121}{16} + 4} = \sqrt{\frac{121 + 64}{16}} = \sqrt{\frac{185}{16}} ]

Finally, substituting the center and radius into the equation of the circle, we get: [ (x + \frac{7}{4})^2 + (y - 2)^2 = (\sqrt{\frac{185}{16}})^2 ]

Therefore, the equation of the circle is ((x + \frac{7}{4})^2 + (y - 2)^2 = \frac{185}{16}).

By signing up, you agree to our Terms of Service and Privacy Policy

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.

- A triangle has corners at #(5 ,8 )#, #(2 ,9 )#, and #(7 ,3 )#. What is the area of the triangle's circumscribed circle?
- A triangle has sides with lengths of 1, 4, and 2. What is the radius of the triangles inscribed circle?
- A triangle has sides with lengths of 5, 9, and 4. What is the radius of the triangles inscribed circle?
- A circle has a chord that goes from #pi/4 # to #pi/8 # radians on the circle. If the area of the circle is #48 pi #, what is the length of the chord?
- Show that #CM# and #RQ# are perpendicular ?

- 98% accuracy study help
- Covers math, physics, chemistry, biology, and more
- Step-by-step, in-depth guides
- Readily available 24/7