Question

The line with equation 2x + y - 5 = 0 is a tangent to the circle with equation (x-3)² (y-p)² = 5. Write down the coordinates of the center of the circle in each case.

Answer 1

To find the coordinates of the center of the circle, you can rearrange the equation of the line in the standard form of a line equation, substitute it into the equation of the circle, and solve for the unknown variable. In this case, the center of the circle is located at (3, 8.125).

Step-by-step explanation:

To find the coordinates of the center of the circle, we can start by rearranging the equation of the line in the standard form of a line equation, which is y = mx + c.

In this case, the equation 2x + y - 5 = 0 can be rearranged to y = -2x + 5.

Next, we substitute this equation into the equation of the circle to find the coordinates of the center.

(x-3)² (y-p)² = 5

Replacing y with -2x + 5, we get:

(x-3)² (-2x + 5 - p)² = 5

Since the line is a tangent to the circle, the discriminant of this equation should be equal to 0.

By expanding and simplifying, we have:

4x² - 20x + 25 - 4px + 6x² - 30x + 25 = 5

Combining like terms, we get:

10x² - 50x + 45 - 4px = 0

For the discriminant to be 0, we have:

(-50)² - 4 * 10 * (45 - 4p) = 0

Simplifying further, we obtain:

2500 - 40(45 - 4p) = 0

Expanding and simplifying, we have:

2500 - 1800 + 160p = 0

Combining like terms, we get:

160p = 1300

Solving for p, we find:

p = 1300/160 = 8.125

Therefore, the center of the circle has coordinates (3, 8.125).