Find the equation for the following circle:

The circle that passes through the origin and has intercepts equal to 1 and 2 on the x- and y-axes, respectively.

That means it must pass though (0,0), (1,0) and (0,2) . The center must be located where the perpendicular bisectors of chords between those points intersect. That would be the point where the vertical line y = 1 and the horizontal ine x= 1/2 intersect, They cross at (1/2, 1) The radius is the distance of that point from any of the three given point, which is

R = sqrt[(1/2)^2 + 1^2] = sqrt (5/4)

Thus the circle equation is
(x-0.5)^2 + (y-1)^2 = 5/4

Check my thinking

Why did the circle go to the therapist?

Because it had issues with its intercepts!

Now, let's find the equation of this troubled circle. We know that the circle passes through the origin, so the center is (0, 0). The intercepts on the x- and y-axes are 1 and 2, respectively.

Using this information, we can determine the equation of the circle as (x - 0)^2 + (y - 0)^2 = r^2, where r is the radius of the circle.

Now, let's find the radius. Since the x-intercept is 1, it means that (1, 0) lies on the circle. Similarly, since the y-intercept is 2, (0, 2) is also on the circle.

Using the distance formula, we can find the distance between (1, 0) and the center (0, 0), which is the radius. So, the radius is sqrt((1-0)^2 + (0-0)^2) = 1.

Therefore, the equation of the circle is:

x^2 + y^2 = 1^2

Simplifying that, we get the final answer:

x^2 + y^2 = 1

To find the equation of a circle, we need to know its center (h, k) and radius r. In this case, the circle passes through the origin, so the center is at (0, 0).

Since the circle has intercepts equal to 1 and 2 on the x- and y-axes respectively, this means the distance from the center to each of the intercepts is equal to the radius.

Let's find the radius first. The distance from the origin to the intercept on the x-axis is 1, and the distance from the origin to the intercept on the y-axis is 2. Using the distance formula, we have:

radius = √[(x2 - x1)^2 + (y2 - y1)^2]
radius = √[(0 - 1)^2 + (0 - 2)^2]
radius = √[1 + 4]
radius = √5

Thus, the radius of the circle is √5.

Now, we can write the equation of the circle in standard form as:

(x - h)^2 + (y - k)^2 = r^2

Substituting the values of the center (h, k) as (0, 0) and the radius r as √5:

(x - 0)^2 + (y - 0)^2 = (√5)^2
x^2 + y^2 = 5

Therefore, the equation of the circle is x^2 + y^2 = 5.

To find the equation of the circle that passes through the origin and has intercepts equal to 1 and 2 on the x- and y-axes, we can use the standard form equation of a circle which is given by:

(x - h)^2 + (y - k)^2 = r^2

where (h, k) represents the coordinates of the center of the circle, and r represents the radius.

Since the circle passes through the origin, the center of the circle will coincide with the origin, which means h = k = 0.

Now, we need to determine the radius of the circle. The radius will be equal to half the length of any of the intercepts. In this case, we can use the intercept on the x-axis, which has a length of 1. So, r = 1/2.

Substituting the values of h, k, and r into the equation, we get:

(x - 0)^2 + (y - 0)^2 = (1/2)^2
x^2 + y^2 = 1/4

Therefore, the equation of the circle is x^2 + y^2 = 1/4.

y=53