Question

In the diagram, a circle centered at the origin, a right triangle, and the Pythagorean theorem are used to derive the equation of a circle, x2 + y2 = r2. If the center of the circle were moved from the origin to the point (h, k) and point P at (x, y) remains on the edge of the circle, which could represent the equation of the new circle? (h + x)2 + (k + y)2 = r2 (x – h)2 + (y – k)2 = r2 (k + x)2 + (h + y)2 = r2 (x – k)2 + (y – h)2 = r2

Answer 1

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

Step-by-step explanation:

Here, the given equation of the circle,

$x^2+y^2=r^2$

Where r be the radius of the circle.

Since, after moving the circle the point P(x,y) is still present in the edge of the new circle,

Thus, the radius of the new circle will be also r.

Also, the radius be the distance from the center to a point on the edge of the circle,

The center of the new circle = (h,k)

⇒ The distance between the points (h,k) and (x,y) = r

⇒ $\sqrt{(x-h)^2+(y-k)^2} = r$

⇒  $(x-h)^2+(y-k)^2= r^2$

Since, (x,y) general representation of the points of the circle,

Hence, this is the required equation of the new circle.

Answer 2

(x – h)2 + (y – k)2 = r2
This is the formula of a circle. The h represents the x value of the center and the k represents the y value of the origin.