Find the equation of the circle concentric with the circle x² + y² - 8x + 14y + 1 = 0 and has half of its area.

Steps to Solve

1. Identify the given circle's parameters

The given circle's equation is ( x^2 + y^2 - 8x + 14y + 1 = 0 ).
To find its center and radius, we will rewrite it in standard form:

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

2. Complete the square

For the ( x ) terms:

$$ x^2 - 8x \rightarrow (x - 4)^2 - 16 $$

For the ( y ) terms:

$$ y^2 + 14y \rightarrow (y + 7)^2 - 49 $$

Thus, the equation can be rewritten as:

$$ (x - 4)^2 + (y + 7)^2 - 16 - 49 + 1 = 0 $$
Simplifying gives:

$$ (x - 4)^2 + (y + 7)^2 = 64 $$

3. Determine the center and radius

From the rewritten equation, we see the center ((4, -7)) and radius ( r = \sqrt{64} = 8 ).

4. Identify properties of the new circle

The new circle is concentric with the given circle, meaning it has the same center ((4, -7)) and must touch the y-axis. The distance from the center to the y-axis is the absolute value of the x-coordinate:

$$ |4| = 4 $$

This is the radius of the new circle.

5. Calculate area and adjust radius

The area of the new circle will be half that of the original circle's area:

Original area:
$$ A_{original} = \pi r^2 = \pi \times 8^2 = 64\pi $$
Thus,
$$ A_{new} = 32\pi $$
So:
$$ \pi r_{new}^2 = 32\pi $$
Thus:
$$ r_{new}^2 = 32 \Rightarrow r_{new} = \sqrt{32} = 4\sqrt{2} $$

6. Write the equation of the new circle

Using the new radius and the same center, the equation of the new circle is:

$$ (x - 4)^2 + (y + 7)^2 = (4\sqrt{2})^2 $$
Simplifying gives:

$$ (x - 4)^2 + (y + 7)^2 = 32 $$