Find all possible radii of circle centered at (3,6) so that the circle intersects only one axis
The distance of the centre from the y-axis is 3, and that from the x-axis is 6.
We therefore require that the radius be more than 3 and up to 6, assuming the interpretation of "intersection" excludes tangency.
so 3<r≤6
To find all possible radii of a circle centered at (3, 6) such that the circle intersects only one axis, we need to consider two cases: when the circle intersects the x-axis and when the circle intersects the y-axis.
1. Circle intersects the x-axis:
When the circle intersects the x-axis, the distance between the center of the circle (3, 6) and any point on the x-axis will be equal to the radius.
Let's assume the radius of the circle is 'r'. In this case, the coordinates of a point on the x-axis can be written as (x, 0), where 'x' is the x-coordinate of the point.
By using the distance formula, we can calculate the distance between the center of the circle (3, 6) and the point (x, 0). The distance is given by:
√((x - 3)² + (0 - 6)²) = r
Simplifying this equation, we get:
(x - 3)² + 36 = r²
Now, since the circle intersects only one axis, we can substitute x = 0 into this equation and solve for r. This will give us the possible radii when the circle intersects the x-axis.
Substituting x = 0, we get:
(0 - 3)² + 36 = r²
9 + 36 = r²
45 = r²
Taking the square root of both sides, we find:
r = √45 = 3√5
Therefore, when the circle intersects the x-axis, the possible radius is 3√5.
2. Circle intersects the y-axis:
Similarly, when the circle intersects the y-axis, the distance between the center of the circle (3, 6) and any point on the y-axis will be equal to the radius.
Assuming the radius of the circle is 'r', the coordinates of a point on the y-axis can be written as (0, y), where 'y' is the y-coordinate of the point.
Using the distance formula, we can calculate the distance between the center of the circle (3, 6) and the point (0, y). The distance is given by:
√((0 - 3)² + (y - 6)²) = r
Simplifying this equation, we have:
9 + (y - 6)² = r²
Again, since the circle intersects only one axis, substitute y = 0 into this equation and solve for r to find the possible radii when the circle intersects the y-axis.
Substituting y = 0, we get:
9 + (0 - 6)² = r²
9 + 36 = r²
45 = r²
Taking the square root of both sides, we find:
r = √45 = 3√5
Therefore, when the circle intersects the y-axis, the possible radius is also 3√5.
Hence, the possible radii of a circle centered at (3, 6) and intersecting only one axis are 3√5.