Problem 58 - Calculus for Biology and Medicine

Question

(a) Find all possible radii of a circle centered at \((2,-5)\) so that the circle intersects only one axis. (b) Find all possible radii of a circle centered at \((2,-5)\) so that the circle intersects both axes.

Step-by-Step Solution

Answer

(a) Radii: 2 or 5. (b) Radius: greater than 5.

Step 1: Understanding the Problem

We need to examine the conditions under which a circle centered at \((2,-5)\) intersects either one axis or both axes. Our center point is \((2, -5)\), where the x-coordinate is 2 and the y-coordinate is -5. Consider the properties of a circle given by the general equation \((x-h)^2 + (y-k)^2 = r^2\), where \((h,k)\) is the center and \(r\) is the radius.

Step 1: Determine Intersection with One Axis

For the circle to intersect only one axis: 1. It should just touch one axis, meaning it is tangent to it, or it can just miss the other. 2. For y-axis intersection only: \(r = 2\) as this would only touch the y-axis at \(x=0\).3. For x-axis intersection only: \(r = 5\) because it's tangent at \(y=0\).So, a radius of either \(r = 2\) or \(r = 5\) means the circle touches only one of the axes.

Step 2: Determine Intersection with Both Axes

The circle must properly intersect both axes. 1. It should pass through both axes rather than be tangent either only to one or neither.2. A circle with \(r > 2\) will intersect the y-axis since it will go into the negative x-direction as the center is at \(x=2\).3. A radius of \(r > 5\) will intersect the x-axis since the center is at \(y=-5\).Thus, the circle must have \(r > 5\) to intersect both axes.

Key Concepts

Circle Equation

The equation of a circle is fundamental in understanding circle geometry. It is given by the formula \((x-h)^2 + (y-k)^2 = r^2\), where \((h, k)\) represents the center of the circle and \(r\) is the radius. This equation encapsulates the set of all points that are equidistant from the circle's center.

This characteristic distance is what we refer to as the radius, a critical component in any circle's geometry. The value of \(r^2\) is crucial because it directly determines the size of the circle. Larger values of \(r\) produce a bigger circle. Conversely, smaller values result in a smaller circle.

• The center \((h, k)\) can be any point on the coordinate plane.
• Every circle has a unique center and radius.

Understanding the circle equation is the first step in solving problems about intersections and tangents.

Intersection with Axes

When determining whether a circle intersects with the axes, we examine its radius relative to its center's distance from the axes. This helps to visualize where and how often the circle may touch or cross the x-axis and y-axis.

If a circle with center at \((2,-5)\) is to intersect only one axis, it must meet specific conditions. For instance, if it is tangent to the **y-axis**, the radius \(r\) must be exactly equal to the distance from the center to the y-axis, which is \(2\). Similarly, to be tangent to the **x-axis**, the radius must be equal to \(5\), matching the distance from the center to the x-axis.

• If \(r = 2\), the circle touches only the y-axis.
• If \(r = 5\), it only touches the x-axis.

Whereas, if a circle intersects both axes, the radius must exceed both distances, so any \(r > 5\) would ensure the circle crosses both axes.

Radius Calculation

Calculating the radius for different intersection scenarios provides insight into the circle's behavior on the coordinate plane. For a circle with a center at \((2, -5)\), determining the correct radius can allow for specialized intersections with the axes.

To calculate when the circle will intersect **both** axes, consider:

• To intersect the **y-axis**, the circle's radius needs to be greater than \(2\).
• To intersect the **x-axis**, the radius must be greater than \(5\).

Thus, the circle must have a radius \(r > 5\) to cover both distances, ensuring it intersects both axes at least once.

Radius calculation is essential for understanding how changes in size affect a circle's interaction with its environment. Whether it touches one axis, both, or none, each scenario hinges on choosing the right value for \(r\).