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?

A: (h + x)2 + (k + y)2 = r2
B: (x – h)2 + (y – k)2 = r2
C: (k + x)2 + (h + y)2 = r2
D: (x – k)2 + (y – h)2 = r2

Answer 1

Option B

Explanation:

Then we have to use distance formula to derive it

• √(x-h)²+(y-k)²=r
• (x-h)²+(y-k)²=r²

Option B is correct

Answer 2

B

Explanation:

`\textsf{Let}\:(x_1,y_1)=(h,k)` → new center of the circle

`\textsf{Let}\:(x_2,y_2)=(x,y)` → point on the circle

Therefore, the radius (r) is the distance between (x, y) and (h, k)

Use the distance formula:

`√((x_2-x_1)^2+(y_2-y_1)^2)=d`

`\implies √((x-h)^2+(y-k)^2)=r`

Square both sides:

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