1) Circle centre (7,-2) passes through (-2,10). Calculate radius and find equation of circle.
2) AB is diameter of circle A(9,5) B(-1,-7). Find equation of circle mid point.
I know i have to use the eqn (x1+x2)/2 and (y1+y2)/2
3) Find the equation of circle with centre (-8,5). Which has y axis as tangent. Find equation of circle.
?? (x+8)²+(y-5)²=r² ??
Please help!
1)
The radius is the distance from the center , (7,-2), to the point (-2,10)
I bet you know the distance formula:
r^2 = (X2 - X1)^2 + (Y2 - Y1)^2
Then
(x-7)^2 +(y+2)^2 = r^2
2)
Yes, the center is at the average of those two points on opposite sides of the circle.
I think it is also asking for the equation of the circle.
Once you have found the center, you can find the equation the same way you found it in #1 but for r^2 use the distance^2 from center to one of those given points.
3)
The y coordinate of the center is 5.
Therefore the tangent to the y axis will be 8 units to the right of the center and at y = 5. (Draw a sketch) In other words at (0,5)
The radius is therefore 8 and r^2 = 64
1) To find the radius and equation of a circle with a given center and a point it passes through, you can use the distance formula between two points. The formula for the distance, d, between two points (x1, y1) and (x2, y2) is:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, the center of the circle is (7, -2), and it passes through (-2, 10). So, the distance between these two points is:
d = sqrt((-2 - 7)^2 + (10 - (-2))^2)
= sqrt((-9)^2 + (12)^2)
= sqrt(81 + 144)
= sqrt(225)
= 15
Therefore, the radius of the circle is 15.
The equation of a circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values of the center and radius, we get the equation of the circle:
(x - 7)^2 + (y + 2)^2 = 15^2
(x - 7)^2 + (y + 2)^2 = 225
2) Since AB is the diameter of the circle, its midpoint will be the center of the circle. To find the midpoint, you can use the midpoint formula:
Midpoint (h, k) = ((x1 + x2)/2, (y1 + y2)/2)
In this case, the coordinates of A are (9, 5) and the coordinates of B are (-1, -7). Plugging these values into the midpoint formula, we get:
Midpoint (h, k) = ((9 + (-1))/2, (5 + (-7))/2)
Midpoint (h, k) = (8/2, -2/2)
Midpoint (h, k) = (4, -1)
So, the coordinates of the midpoint are (4, -1), which is also the center of the circle.
The equation of the circle with center (h, k) and radius r is:
(x - h)^2 + (y - k)^2 = r^2
Plugging in the values of the center and radius, we get the equation of the circle:
(x - 4)^2 + (y + 1)^2 = r^2
3) The equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
In this case, the center of the circle is (-8, 5), and the y-axis is tangent to the circle. This means that the distance between the center of the circle and the y-axis is equal to the radius of the circle.
Since the distance between the center of the circle and the y-axis is the x-coordinate of the center (which is -8), the radius of the circle is 8.
Plugging in the values of the center and radius into the equation of a circle, we get:
(x - (-8))^2 + (y - 5)^2 = 8^2
(x + 8)^2 + (y - 5)^2 = 64
Therefore, the equation of the circle is (x + 8)^2 + (y - 5)^2 = 64.