algebra 2 question?

Question

how would you find the standard form of an equation of a circle if only given the end points of a diameter?

endpoints of a diameter:(0,0) (6,8)

Answer 1

Recall that the equation for a circle is (x-h)^2 + (y-k)^2 = r^2 where (h,k) is the center of the circle and r is the radius. (It helps to draw a quick sketch of the circle in your coordinate system.)
The radius is easy to find. Use the pythagorean theorem to find the length of the diameter. Then divide it in half. so...
a^2 + b^2 = c^2
get a from the x coordinates: 6-0 = 6
and b from the y coordinates: 8-0 = 8
6^2 + 8^2 = c^2
36+64=100=c^2
c=sqrt(100)
c=10
diameter = 10
radius = diameter/2
radius = 10/2 = 5

Now you need to go back to the circle and look and see if you can tell the coordinates of the center of the circle, (h,k).
The radius is half of the diameter and the coordinates of one of its endpoints (the one at the center of the circle) are halfway between the coordinates that gave the diameter. Half of the difference between the x coordinates (0 and 6) is 3. Half of the difference between the y coordinates ( 0 and 8) is 4. So the center of the circle (h,k) is located at (3,4). Now just plug h,k, and r into the circle equation to get
((x-3)^2)+((y-4)^2)=25

That was a little involved wasn't it? ^.^

In case you are just totally lost (i don't blame you) here's a picture: http://www.4freeimagehost.com/show.php?i=e4e632645a58.jpg

Answer 2

The center of the circle is the midpoint of the line segment making the diameter AB.
The midpoint formula is used to find the coordinates of the center C of the circle.
x coordinate of C = (0 + 6) /2 = 3
y coordinate of C = (0 + 8) / 2 = 4

The radius is half the distance between A and B.
r = (1/2) ([8 - 0]^2 + [6 - 0]^2 )^1/2
= (1/2)(8^2 + 6^2)^1/2
= (1/2)(8 + 6)
= (1/2)(14)
= 7

The coordinate of C and the radius are used in the standard equation of the circle to obtain the equation:
(x - 3)^2 + (y - 4)^2 = 7^2
(x - 3)^2 + (y - 4)^2 = 49

Answer 3

The standard form of a circle is (x-h)^2+(y-k)^2=r^2

(h, k) is the center
r is the radius

Given the two points, the midpoint of the line segment connecting them is the center of the circle. The length of the segment is the diameter. So, the radius is the length divided by two.

To find the midpoint, use the Midpoint Formula:

___(x1+x2__y1+y2)
M=(--------- + ---------)
___(_2______2___) sorry formatting is screwed up.

so M= ( (0+6)/2, (0+8)/2)=(3, 4)

To find the distance, use the Distance Formula:

d= sqrt( (y2-y1)^2 + (x2-x1)^2 )

d=sqrt ( (8-0)^2 + (6-0)^2 ) = sqrt(64+36)=sqrt(100)=10

diameter= 10
so, radius=5

so the circle is: (x-3)^2+(y-4)^2=25



ALTERNATIVELY, you can notice that the (0,0), (6,0), and (6,8) points make a 6-8-10 triangle (diameter=10) with the midpoint of the line segment occuring at (3,4) by the Mid-Line Theorem. <-----much easier

Answer 4

endpoints of a diameter:(0,0) (6,8)
So, the center is (3,4)
so the radius is r = sqrt.(3^2 + 4^2) = sqrt.(25) = 5

Remember that a circle is a locus of points. A circle is all of the points that are a fixed distance, known as the radius, from a given point, known as the center of the circle.
On the coordiante plane, the formula becomes
(X-H)^2 + (Y-K)^2 = r^2
h and k are the x and y coordinates of the center of the circle
(x - 3)^2 + (y - 4)^2 = 25

Answer 5

As we have the coordinates of the end-points of the diameter, then we can easily find the coordinate of the center of the circle, which is the mid-point of the two end-points.

coordinate of the center is ( (0+6)/2, (0+8)/2 ) or, (3,4)

Length of the radius
= √(3-0)^2+(4-0)^2
= √(9+16)
= √25
= ±5
= 5 [Negative is avoided, as length cannot be negative]

So, the equation of the circle with center at (3,4) and radius 5 is as follows:

(x-3)^2 + (y-4)^2 = (5)^2
or, (x-3)^2 + (y-4)^2 = 25 [Answer]

Answer 6

The midpoint of the circle will be:

(x1 + x2)/2 and (y1 + y1)/2

Therefore, substituting the values you have gives

(6 + 0)/2 = 3 and (8 + 0)/2 = 4

Therefore, the midpoint of the circle is (3,4)

let's call 3 "h" and 4 "k"

The radius of the circle found by using the distance formula and dividing it in half. The distance formula for the points is:

square root( (x2 -x1)^2 + (y2 - y1)^2)
square root( (6 - 0)^2 + (8 - 0)^2)
square root(36 + 64)
square root (100)
square root(100) = 10

Since the radius is half the diameter, the radius is 10/2 = 5

The standard equation for a circle is:

(x - h)^2 + (y - k)^2 = r^2 where r is the radius.

Therefore, the standard equation of this circle is:

(x - 3)^2 + (y - 4)^2 = 5^2

Answer 7

if the point :(0,0) (6,8) are the ends of its diameter, then the midpoint of those point is the center of the circle

let x1 and y1 be (0,0) x2 and y2 be (6,8)
the midpoint formula is = x2 -x1/2 and y2 - y1/2
by using the formula we get
6 - 0/2 = 3
x = 3
8 - 0/2 = 4
y = 4
so the center is (3,4)
the radius is half the length of the diameter, so we will find the diameter

by using the distance formula
\/(x2-x1)^2+(y2-y1)^2
= \/(6- 0)^2 + (8-0)^2
= \/36 + 64
= \/100
= 10
so the diameter is 10 and half of it is the radius which is 5

the standard form if the circle is (x-h)^2 + (y - k)^ = r^2
where h and k is the center and r is the radius, so we plug in those values
(x-3)^ +(y-4)^2 = 5^2
the final answer is
(x-3)^2 +(y-4)^2 = 25

thats all!

Answer 8

From the given coordinates you can find the center of the circle which would be (3, 4). you can also find the radius from the given coordinates as well which is half of the distance between the two pints which is 10 /2 =5. Having the center of the circle and it's radius as above, you have the equation of that circle: (x-3)^2+(y-4)^2=25. That's all.

As an exercise, try to think of the details of my calculations

Answer 9

centre (3 , 4)
radius = 5
(x - 3) ² + (y - 4) ² = 5 ²
(x - 3) ² + (y - 4) ² = 25