Problem 61 - Precalculus: Mathematics for Calculus

Question

The ancillary circle of an ellipse is the circle with radius equal to half the length of the minor axis and center the same as the ellipse (see the figure). The ancillary circle is thus the largest circle that can fit within an ellipse. (a) Find an equation for the ancillary circle of the ellipse $$x^{2}+4 y^{2}=16$$ (b) For the ellipse and ancillary circle of part (a), show that if \((s, t)\) is a point on the ancillary circle, then \((2 s, t)\) is a point on the ellipse. (IMAGE CAN'T COPY)

Step-by-Step Solution

Answer

(a) The ancillary circle's equation is \( x^2 + y^2 = 4 \). (b) For point \((s, t)\) on the circle, \((2s, t)\) is on the ellipse.

Step 1: Understand the equation of the ellipse

The given ellipse equation is \( x^2 + 4y^2 = 16 \). We can write it in standard form as \( \frac{x^2}{16} + \frac{y^2}{4} = 1 \). This reveals that the semi-major axis (\( a \)) is 4 and the semi-minor axis (\( b \)) is 2.

Step 2: Find the radius of the ancillary circle

The ancillary circle has a radius equal to the semi-minor axis, which is \( b = 2 \). This circle is centered at the same point as the ellipse, which is the origin \((0, 0)\).

Step 3: Write the equation for the ancillary circle

Since the circle is centered at the origin and has a radius of 2, its equation will be \( x^2 + y^2 = 4 \).

Step 4: Show the relation between points on the ancillary circle and the ellipse

Consider a point \((s, t)\) on the ancillary circle where \( s^2 + t^2 = 4 \). If we scale this point by replacing \(s\) with \(2s\), then \((2s, t)\), it becomes a point on the ellipse. Substitute into the ellipse equation: \( (2s)^2 + 4t^2 = 4s^2 + 4t^2 = 4(s^2 + t^2) = 4 \times 4 = 16 \), satisfying the original ellipse equation \( x^2 + 4y^2 = 16 \). Thus, \((2s, t)\) lies on the ellipse.

Key Concepts

Ellipse Equation

An ellipse is a geometric shape that appears like a flattened circle. The standard equation of an ellipse can be expressed as \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\), where \(a\) and \(b\) represent the lengths of the semi-major and semi-minor axes, respectively. These axes help determine the shape and orientation of the ellipse.
In the equation provided, \(x^2 + 4y^2 = 16\), we rearrange it to the standard form: \(\frac{x^2}{16} + \frac{y^2}{4} = 1\).
Identifying the values gives us:

• Semi-major axis, \(a = 4\)
• Semi-minor axis, \(b = 2\)

The larger value, \(a\), indicates the direction in which the ellipse is stretched. Here, \(a\) is aligned along the x-axis, making it the major axis.

Equation of a Circle

The equation for a circle is a simple formula that reveals its symmetrical nature. A standard circle equation centered at the origin is \(x^2 + y^2 = r^2\), where \(r\) is the radius. This equation signifies that every point \((x, y)\) on the circle is at an equal distance \(r\) from the center \((0,0)\).
In the context of an auxiliary circle related to the ellipse \(x^2 + 4y^2 = 16\), the radius is equal to the semi-minor axis of the ellipse, \(r = 2\).
Thus, the equation of the ancillary circle becomes:

• Equation: \(x^2 + y^2 = 4\)

This signifies that any point on this circle is 2 units away from the origin, matching the semi-minor axis length of the ellipse.

Ellipse and Circle Relationship

The relationship between an ellipse and its ancillary circle can help understand how these shapes connect and interact.
The ancillary circle of an ellipse is the largest circle that can fit within it, having its radius as the semi-minor axis \(b\) of the ellipse. This circle shares the same center as the ellipse.
This relationship is useful to explore point correspondences. For instance, given point \((s, t)\) on the ancillary circle with \(s^2 + t^2 = 4\), we can see how this transforms to a point on the ellipse. By scaling \(s\) to \(2s\), the point \((2s, t)\) meets the condition of the ellipse equation \(x^2 + 4y^2 = 16\):

• If \((2s)^2 + 4t^2 = 16\) holds, then \((2s, t)\) is on the ellipse.

This demonstrates how every point on the ancillary circle can be scaled to become a point on the ellipse, linking the two together geometrically.