Problem 58 - Calculus for Biology and Medicine

Question

(a) Find all possible radii of a circle centered at \((3,6)\) so that the circle intersects only one axis. (b) Find all possible radii of a circle centered at \((3,6)\) so that the circle intersects both axes.

Step-by-Step Solution

Answer

(a) Radii = 3 or 6. (b) Radii > 6 and ≤ 3√5.

Step 1: Understand the Circle Equation

The equation of a circle centered at a point \( (h, k) \) with radius \( r \) is given by \( (x-h)^2 + (y-k)^2 = r^2 \). For our circle, it is centered at \( (3, 6) \), so the equation becomes \( (x-3)^2 + (y-6)^2 = r^2 \).

Step 2: Circle Intersecting Only One Axis

For the circle to intersect only one axis, it must touch the x-axis or the y-axis at exactly one point. 1. **X-axis intersection:** The distance from the circle's center to the x-axis should equal the radius, so \( k = r \) for the x-axis. Thus, \( r = 6 \).2. **Y-axis intersection:** Similarly, the distance from the center to the y-axis should equal the radius, which gives \( h = r \). Thus, \( r = 3 \).For the circle to intersect one axis, the possible radii are only 3 or 6.

Step 3: Circle Intersecting Both Axes

For the circle to intersect both axes, the radius must be large enough to reach both the x-axis and y-axis, but not so large that it completely passes over any axis.1. **Minimum radius:** Should reach both axes so \( r > 6 \), because distance from center to the x-axis is 6.2. **Maximum radius:** Must not exceed the diagonal distance from the center point (3,6) to the origin (0,0), calculated using the Pythagorean theorem, so the maximum value \( r \) is \( \sqrt{3^2 + 6^2} = \sqrt{45} = 3\sqrt{5} \).3. So, the possible radii that allow the circle to intersect both axes are \( 6 < r \leq 3\sqrt{5} \).

Key Concepts

Intersection of a Circle with Axes

A circle intersects an axis when it touches or crosses it. When dealing with a circle centered at a point like \(3, 6\), determining intersections with the x-axis and y-axis involves a few interesting scenarios.
To simplify:

• For a circle to intersect only one axis, it must touch but not cross that axis.
• For it to intersect both axes, it must cross them at least once.

If a circle intersects just one axis, it remains tangent to that axis, touching it at exactly one point. When looking for situations where a circle intersects both axes, we need a radius large enough to connect to both but not excessively big, which would make it pass over and no longer intersect the x or y-axis.
Analyzing these conditions requires careful calculation of distances from the center to each axis.

Identifying the Circle Center

The location of the circle's center \(h, k\) plays a crucial role in determining intersection points. Here, the circle is centered at \(3, 6\). This means:

• Its distance from the x-axis (horizontal line) is the y-coordinate, 6 units above the origin.
• Its distance from the y-axis (vertical line) is the x-coordinate, 3 units from the origin.

Understanding these distances helps in visualizing how and where the circle will intersect the axes.
In essence, the center dictates the starting point. By calculating radii relative to these distances, you determine if the circle reaches (intersects) or falls short of the axes. The closer the center is to an axis, the smaller the radius needed to achieve intersection.

Understanding the Radius of a Circle

The radius is the distance from the center of the circle to any point on its boundary. It determines the size of the circle and directly affects intersection points with axes.
For the given center \(3, 6\):

• To intersect only the x-axis, the radius must be exactly 6, matching the vertical distance from the center to the x-axis.
• To intersect only the y-axis, the radius must be exactly 3, matching the horizontal distance from the center to the y-axis.
• For the circle to hit both axes, the radius must be larger than 6 but not exceed \3\sqrt{5}\. This ensures it stretches enough to reach each axis without completely bypassing them.

The balance and proper calculation of the radius ensure the circle behaves as desired, meeting the conditions for single or dual axis intersections.