Question

X² + y² = 25; left 3 units and down 2 units

Answer 1

If you want to shift the circle defined by \(x^2 + y^2 = 25\) three units to the left and two units down, you can achieve this by adjusting the coordinates in the equation. The original equation represents a circle centered at the origin (0,0) with a radius of 5.

Shift to the left by 3 units means subtracting 3 from the x-coordinate, and shifting down by 2 units means subtracting 2 from the y-coordinate. So, the modified equation would be:

\[(x + 3)^2 + (y - 2)^2 = 25\]

This equation represents a circle with its center at (-3, 2), three units to the left and two units down from the original center, and the radius remains 5.

Answer 2

The equation x² + y² = 25 is for a circle centered at the origin with a radius of 5. To move it left by 3 units and down by 2 units, the equation transforms to (x + 3)² + (y + 2)² = 25, indicating a shift in the center of the circle to the point (-3, -2).

Step-by-step explanation:

The question involves applying transformations to a circle's equation in a coordinate plane. The original equation x² + y² = 25 represents a circle with a radius of 5 units, centered at the origin (0,0). To move this circle left by 3 units and down by 2 units, we adjust the equation to account for the horizontal and vertical shifts.

Shifting the circle left by 3 units changes the x-coordinate of the center, and shifting it down by 2 units changes the y-coordinate of the center. The transformed equation becomes (x + 3)² + (y + 2)² = 25. This new equation now describes a circle with the same radius of 5 units but its center is now located at (-3, -2), which is 3 units to the left and 2 units down from the original center.

The principle used here is related to the Pythagorean theorem, which can explain the relationship between the coordinates of a point on the circumference of the circle and the radius. The shifting of the circle is an example of how geometric transformations are applied in algebraic forms.