Question

Use Fermat's method to determine the subtangent to the ellipse \(x^{2} / a^{2}+y^{2} / b^{2}=1\). Compare your answer with that of Apollonius in Chapter 4 .

Short answer

Question: Determine the subtangent of the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\) using Fermat's method and compare the result with Apollonius' result in Chapter 4. Answer: The subtangent of the given ellipse is \(\frac{a^2y}{b^2}\), which is the same as Apollonius' result, \(\frac{a^2}{b} \cdot y\).

Step-by-step solution

Write the ellipse equation

Given ellipse equation: \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\)

Rewrite the ellipse equation as a function of x and y

Rewrite the given ellipse equation as a function of x and y: \(y^2 = b^2\left(1 - \frac{x^2}{a^2}\right)\)

Find the derivative of y with respect to x

Differentiating both sides with respect to x, using the chain rule, we get: \(\frac{d y^2}{d x} = -2 \cdot b^2 \cdot \frac{x}{a^2} \cdot \frac{d y}{d x}\) As \(y(t) = \sqrt{b^2(1 - \frac{x^2}{a^2})}\), we can obtain the derivative of y with respect to x by differentiating this expression: \(\frac{dy}{dx} = \frac{-(b^2)x}{a^2y}\)

Apply Fermat's theorem on the subtangent

Fermat's theorem states that if the tangent to a curve at a point intersects the x-axis, the subtangent is equal to the x-coordinate of the point divided by the slope of the tangent. Using Fermat's theorem and the derivative we found in Step 3: Subtangent = \(\frac{x}{\frac{-b^2x}{a^2y}} = \frac{a^2y}{b^2}\)

Compare the answer with Apollonius' result in Chapter 4

Apollonius' result in Chapter 4 is: Subtangent = \(\frac{a^2}{b} \cdot y\) Comparing our result with Apollonius', we can see that both answers are essentially the same as we can rewrite our result as \(\frac{a^2}{b} \cdot y\). Therefore, using Fermat's method, we have determined the subtangent to the ellipse and verified that it is the same as the one given by Apollonius.

Key concepts

Subtangent

When dealing with curves such as ellipses, the concept of a "subtangent" is important for understanding how a tangent line intercepts the x-axis. Essentially, the subtangent is part of the horizontal projection of the tangent on the x-axis. It quantifies how far along the x-axis the tangent line extends from a given point of tangency.

To determine the subtangent using Fermat's method, you need to follow the steps given in the mathematical solution process:

• First, express the equation of the ellipse in terms of a function of x and y.
• Second, compute the derivative of this function to find the slope of the tangent line at a specific point.
• Finally, use Fermat's theorem which tells you that the subtangent is the x-coordinate divided by the tangent's slope.

In this case, for an ellipse described by the equation \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \), the subtangent was calculated to be \( \frac{a^2}{b} \cdot y \), which matched the result obtained by Apollonius.Understanding subtangents deeply can help enhance your grasp of geometric properties and analyses pertaining to curves like ellipses.

Ellipse

Ellipses are one of the fascinating and fundamental shapes in geometry and appear prolifically across mathematics and physics. An ellipse is the locus of all points where the sum of the distances from two fixed points—known as the foci—remains constant.

The standard equation for an ellipse centered at the origin is given by \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \). Here:

• \(a\) represents the semi-major axis, the longest radius extending from the center to the edge of the ellipse.
• \(b\) is the semi-minor axis, the shortest radius.

The geometry of ellipses becomes crucial in fields such as astronomy where the path of planets and comets around the sun is elliptical.

By using differentiation on the ellipse’s equation, we are able to gain mathematical insights into how tangent lines behave at any given point, which aids the calculation of subtangents and various other parameters related to ellipses.

Apollonius

Apollonius of Perga was a renowned Greek geometer, particularly famed for his work on conic sections, which includes ellipses, parabolas, and hyperbolas. His results are foundational and continue to influence modern mathematics and geometry.

In the context of our problem involving the ellipse, Apollonius had investigated the properties and characteristics of tangents and subtangents to conic sections centuries before calculus was formally developed. This work in Chapter 4 of his treatises provided insights that enable mathematicians to calculate these properties precisely.

Revisiting Apollonius' work and comparing it with methods like Fermat's, as in the exercise, is critical as it not only showcases the ingenuity of early mathematicians but also allows us to appreciate the mathematical continuity and advancement over the ages. The fact that Fermat's work aligns with that of Apollonius illustrates the robustness and accuracy of classical geometry.