Question

Consider a circle which touches the x-axis at point (3, 0) and passes through point (8, 5). Then the length of the diameter of the circle is

• A. 5 units
• B. 15 units
• C. 20 units
• D. 10 units

Answer 1

Answer: (D)

10 units

Answer 2

Let the center of the circle be (h, k) and its radius be r.

Since the circle touches the x-axis at the point (3, 0), the x-coordinate of the center of the circle must be 3. Thus, h = 3.

The radius of the circle, r, is the perpendicular distance from the center (h, k) to the x-axis, which is |k|.

Given that the circle passes through the point (8, 5), which has a positive y-coordinate, the center's y-coordinate (k) must also be positive. Therefore, r = k.

The equation of the circle is given by $(x - h)^2 + (y - k)^2 = r^2$.

Substituting h = 3 and r = k, the equation becomes:

$(x - 3)^2 + (y - k)^2 = k^2$

The circle passes through the point (8, 5). Substituting x = 8 and y = 5 into the equation:

$(8 - 3)^2 + (5 - k)^2 = k^2$ $5^2 + (5 - k)^2 = k^2$ $25 + (25 - 10k + k^2) = k^2$ $25 + 25 - 10k + k^2 = k^2$ $50 - 10k + k^2 = k^2$

Subtract $k^2$ from both sides:

$50 - 10k = 0$ $10k = 50$ $k = \frac{50}{10}$ $k = 5$

Since r = k, the radius of the circle is r = 5 units.

The length of the diameter of the circle is $2 \times r$.

Diameter = $2 \times 5 = 10$ units.