A circle is tangent to the y-axis at the point (0,2) and passes through the point (8,0). Find the radius of the circle.
I tried using distance formula but it doesnt work? Help please thanks.
since the line y = 2 is tangent to the circle, its centre must lie on the y-axis
let the centre be (0,b)
the equation of the circle is
x^2+ (y-b)^2 = r^2
but (0,2) lies on it ----> (2-b)^2 = r^2
but (8,0) lies on it ----> 64 + (0-b)^2 = r^2
64 +b^2= r^2
then 64 + b^2 = (2-b)^2
64 + b^2 = 4-2b+b^2
60 = -2b
b = -30
the centre is (0, -30) ---> radius = 2-(-30) = 32
OR
distance from (0,b) to (0,2) must equal the distance from (0,2) to (8,0)
2-b = √(64 + b^2)
square both sides
4-2b+b^2 = 64+b^2
-2b = 60
b = -30
the radius is 2 - (-30) = 32
That is super wrong.
To find the radius of the circle, we can use the distance formula between two points on the circle. However, in this case, using the distance formula directly between (0,2) and (8,0) will not give us the radius, as those points are not diametrically opposite on the circle.
Instead, we can use the fact that the center of the circle lies on the perpendicular bisector of the line segment joining (0,2) and (8,0). The point (0,2) is the point of tangency to the y-axis, which means that the center of the circle lies on the line x = 0.
Therefore, we can find the equation of the perpendicular bisector of the line segment joining (0,2) and (8,0), and then find the intersection of that line with the line x = 0 to determine the center of the circle.
The midpoint of the line segment joining (0,2) and (8,0) can be found by taking the average of their x-coordinates and y-coordinates:
Midpoint = ((0+8)/2, (2+0)/2) = (4, 1)
The slope of the line segment joining (0,2) and (8,0) can be found using the slope formula:
Slope = (change in y) / (change in x) = (0 - 2) / (8 - 0) = -2/8 = -1/4
The slope of the perpendicular bisector is the negative reciprocal of the slope of the line segment, so the slope of the perpendicular bisector is 4.
Using the slope-intercept form of a line (y = mx + b), we can substitute the slope (4) and the coordinates of the midpoint (4,1) to find the y-intercept (b):
1 = 4(4) + b
1 = 16 + b
b = 1 - 16
b = -15
So, the equation of the perpendicular bisector of the line segment joining (0,2) and (8,0) is y = 4x - 15.
To find the center of the circle, we need to find the intersection of this line with the line x = 0. Substituting x = 0 into the equation of the perpendicular bisector:
y = 4(0) - 15
y = -15
Therefore, the center of the circle is (0, -15).
To find the radius of the circle, we can use the distance formula between the center of the circle (0, -15) and any point on the circle, such as (8,0):
Distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
= sqrt((8 - 0)^2 + (0 - (-15))^2)
= sqrt(8^2 + 15^2)
= sqrt(64 + 225)
= sqrt(289)
= 17
Hence, the radius of the circle is 17.