I have no idea how to do these problems:
Find an equation of each circle.
1. Center (3,5); tangent to the x axis
2. Center (5,-3); tangent to the y axis
3. Tangent to the x axis, y axis, and the line y=5 (two answers)
I just really need help understanding how to do this.
1. Did you sketch or just visualize the circle?
Isn't the radius 5 ?
so (x-3)^2 + (y-5)^2 = 25
2. Do it the same way.
3. y = 5 is a horizontal line 5 units above the x-axis.
I can visualize a circle in the first quadrant, and another in the second quadrant.
isn't the diameter 5 ?
so isn't the centre 2.5 up from the x-axis and 2.5 to the right of the y-axis?
mmmh?
To find the equation of a circle given its center and any additional conditions, such as being tangent to an axis or a line, you need to understand the general equation of a circle and how to apply the given conditions.
The general equation of a circle with center (h, k) and radius r is given by:
(x - h)^2 + (y - k)^2 = r^2
Now, let's go through each problem step by step:
1. Center (3,5); tangent to the x-axis:
Since the circle is tangent to the x-axis, the distance from the center of the circle to the x-axis will be equal to its radius. In this case, the radius will be the vertical distance from the center to the x-axis, which is |5 - 0| = 5.
Thus, the equation of the circle will be:
(x - 3)^2 + (y - 5)^2 = 5^2
Simplifying, you get:
(x - 3)^2 + (y - 5)^2 = 25
2. Center (5,-3); tangent to the y-axis:
Similar to the previous problem, since the circle is tangent to the y-axis, the distance from the center of the circle to the y-axis will be equal to its radius. In this case, the radius will be the horizontal distance from the center to the y-axis, which is |5 - 0| = 5.
Therefore, the equation of the circle will be:
(x - 5)^2 + (y + 3)^2 = 5^2
Simplifying, you get:
(x - 5)^2 + (y + 3)^2 = 25
3. Tangent to the x-axis, y-axis, and the line y = 5 (two possible answers):
For this problem, the circle is tangent to both the x-axis and the y-axis, meaning its center lies on the line y = x (since the distance from the center to both axes would be the same).
One possible solution is when the center is at (5, 5), where the vertical distance from the center to the x-axis is 5 (equal to the radius). In this case, the equation of the circle will be:
(x - 5)^2 + (y - 5)^2 = 5^2
The second possible solution is when the center is at (-5, 5), where the horizontal distance from the center to the y-axis is 5 (equal to the radius). In this case, the equation of the circle will be:
(x + 5)^2 + (y - 5)^2 = 5^2
Remember to evaluate the distance from the center to determine the radius, and apply the conditions given in the problem to find the equation of the circle.