A circle has a center that falls on the line $y = \frac{5}{2} x + 1$ $y = 5/2x +1$ and passes through $(8, 2)$ $(8 ,2 )$ and $(3, 1)$ $(3 ,1 )$ . What is the equation of the circle?

Question

A circle has a center that falls on the line $y = \frac{5}{2} x + 1$ $y = 5/2x +1$ and passes through $(8, 2)$ $(8 ,2 )$ and $(3, 1)$ $(3 ,1 )$ . What is the equation of the circle?

1 Answer

Impact of this question

1511 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2016-10-21T11:39:32

The equatio of the circle is ${(x - \frac{56}{15})}^{2} + {(y - \frac{31}{3})}^{2} = 87.65$ $(x-56/15)^2+(y-31/3)^2=87.65$

Explanation:

Let the center of the cicle be $(a, b)$ $(a,b)$
As the line passes through the center $b = \frac{5 a}{2} + 1$ $b=(5a)/2+1$
The equation of the circle is ${(x - a)}^{2} + {(y - b)}^{2} = r^{2}$ $(x-a)^2+(y-b)^2=r^2$
where $r$ $r$ is the radius
As the circle passes through $(8, 2)$ $(8,2)$ and (3,1)
we get, ${(8 - a)}^{2} + {(2 - b)}^{2} = r^{2}$ $(8-a)^2+(2-b)^2=r^2$
and ${(3 - a)}^{2} + {(1 - b)}^{2} = r^{2}$ $(3-a)^2+(1-b)^2=r^2$
Equating ${(8 - a)}^{2} + {(2 - b)}^{2} = {(3 - a)}^{2} + {(1 - b)}^{2}$ $(8-a)^2+(2-b)^2=(3-a)^2+(1-b)^2$
Developing
$64 - 16 a + a^{2} + 4 - 4 b + b^{2} = 9 - 6 a + a^{2} + 1 - 2 b + b^{2}$ $64-16a+a^2+4-4b+b^2=9-6a+a^2+1-2b+b^2$

$64 - 16 a + 4 - 4 b = 9 - 6 a - 2 b + 1$ $64-16a+4-4b=9-6a-2b+1$

$68 - 16 a - 4 b = 10 - 6 a - 2 b$ $68-16a-4b=10-6a-2b$

$58 = 10 a + 2 b$ $58=10a+2b$

$5 a + b = 29$ $5a+b=29$ we compare this to the first equation
$b = \frac{5 a}{2} + 1$ $b=(5a)/2+1$
$29 - 5 a = \frac{5 a}{2} + 1$ $29-5a=(5a)/2+1$
$15 \frac{a}{2} = 28$ $15a/2=28$
$a = 2 \cdot \frac{28}{15} = \frac{56}{15}$ $a=2*28/15=56/15$
and $b = \frac{5}{2} \cdot \frac{56}{15} + 1 = \frac{31}{3}$ $b=5/2*56/15+1=31/3$
So the radius is $r^{2} = {(3 - \frac{56}{15})}^{2} + {(1 - \frac{31}{3})}^{2} = \frac{11^{2}}{15^{2}} + \frac{28^{2}}{9} = 87.65$ $r^2=(3-56/15)^2+(1-31/3)^2=11^2/15^2+28^2/9=87.65$
and $r = 9.36$ $r=9.36$

Science

Math

Humanities

... and beyond

A circle has a center that falls on the line $y = \frac{5}{2} x + 1$ $y = 5/2x +1$ and passes through $(8, 2)$ $(8 ,2 )$ and $(3, 1)$ $(3 ,1 )$ . What is the equation of the circle?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

A circle has a center that falls on the line y=52x+1y = 5/2x +1 and passes through (8,2)(8 ,2 ) and (3,1)(3 ,1 ). What is the equation of the circle?

1 Answer

Explanation:

Related questions

Impact of this question

A circle has a center that falls on the line $y = \frac{5}{2} x + 1$ $y = 5/2x +1$ and passes through $(8, 2)$ $(8 ,2 )$ and $(3, 1)$ $(3 ,1 )$ . What is the equation of the circle?