Problem 49 - Calculus Single Variable

Question

The curve \(x^{2}-x y+y^{2}=4\) is an ellipse. Notice that \(P=(a, b)\) is on the ellipse if and only if \(Q=(-a,-b)\) is on the ellipse. Assuming that \(P\) and \(Q\) are both points on the ellipse, show that the tangent lines at \(P\) and \(Q\) are parallel.

Step-by-Step Solution

Answer

Tangent lines at points \((a, b)\) and \((-a, -b)\) are parallel because their slopes are equal.

Step 1: Analyze the Ellipse Equation

Given the equation \(x^2 - xy + y^2 = 4\), notice that this is a quadratic equation in \(x\) and \(y\). The terms \(x^2\), \(-xy\), and \(y^2\) suggest a conic section, and the given information tells us this is an ellipse.

Step 2: Understand the Symmetry Property

The problem states that if \((a, b)\) is on the ellipse, then \((-a,-b)\) is too. This indicates symmetry about the origin, as both points satisfy the equation \(x^2 - xy + y^2 = 4\).

Step 3: Find the Tangent Line Slope at a Point

To find the tangent line at point \((a, b)\), we first perform implicit differentiation of the ellipse equation \(x^2 - xy + y^2 = 4\) with respect to \(x\).

Step 4: Perform Implicit Differentiation

Differentiate both sides of the equation \(x^2 - xy + y^2 = 4\) w.r.t. \(x\):\[\frac{d}{dx}(x^2) - \frac{d}{dx}(xy) + \frac{d}{dx}(y^2) = \frac{d}{dx}(4)\]This yields:\[2x - (y + x \frac{dy}{dx}) + 2y \frac{dy}{dx} = 0\]

Step 5: Solve for \(\frac{dy}{dx}\) (slope)

Rearrange to solve for \(\frac{dy}{dx}\):\[2x - y - x \frac{dy}{dx} + 2y \frac{dy}{dx} = 0\]\[2x - y = \frac{dy}{dx}(x - 2y)\]Thus, the slope at \((x, y)\) is:\[\frac{dy}{dx} = \frac{2x - y}{x - 2y}\]

Step 6: Find the Tangent Slope at \(P=(a, b)\)

Substitute \(x = a\) and \(y = b\) into the slope formula:\[\left.\frac{dy}{dx}\right|_P = \frac{2a - b}{a - 2b}\]

Step 7: Find the Tangent Slope at \(Q=(-a, -b)\)

Substitute \(x = -a\) and \(y = -b\) into the slope formula:\[\left.\frac{dy}{dx}\right|_Q = \frac{-2a + b}{-a + 2b}\]This simplifies to:\[\frac{dy}{dx}|_Q = \frac{2a - b}{a - 2b}\]The expression is equivalent to the slope at \(P\).

Step 8: Conclude Tangent Lines are Parallel

Since the slopes of the tangent lines at both \((a, b)\) and \((-a, -b)\) are the same, the tangent lines are parallel.

Key Concepts

Symmetry in Ellipses

Ellipses are a type of conic section known for their symmetrical properties. Symmetry in an ellipse implies that if there is a point on the ellipse, such as \((a, b)\), then there is another corresponding point \((-a, -b)\) also on the ellipse, given the plane symmetry about the origin.
This symmetry arises from how the standard equation for an ellipse balances the coefficients of the terms. For our specific ellipse represented by the equation \(x^2 - xy + y^2 = 4\), the symmetry is maintained across the origin.
This characteristic is derived from the fact that the ellipse's equation remains unchanged when replacing \(x\) with \(-x\) and \(y\) with \(-y\). This results in the ellipse being symmetric with respect to the origin, which holds true for many conic sections like ellipses, hyperbolas, and circles.

Implicit Differentiation

Implicit differentiation is a useful technique for finding derivatives when dealing with relations where \(y\) is not explicitly expressed as a function of \(x\). It allows us to handle relations like the given ellipse equation: \(x^2 - xy + y^2 = 4\).
To perform implicit differentiation, each term of the equation is differentiated with respect to \(x\), while treating \(y\) as a function of \(x\).

• The term \(x^2\) becomes \(2x\).
• The term \(-xy\) requires the product rule, resulting in \(-(y + x \frac{dy}{dx})\).
• The term \(y^2\) becomes \(2y \frac{dy}{dx}\) as it involves \(y\) implicitly.

Once differentiated, these parts are combined to find \(\frac{dy}{dx}\), giving us the slope formula derivable from the equation. This is crucial for finding tangent lines.

Tangent Line of an Ellipse

A tangent line to a curve at a given point is a line that just "touches" the curve at that point, without crossing it. For finding this line on an ellipse, we need the slope at a specified point.
By using implicit differentiation, we discovered that for our ellipse \(x^2 - xy + y^2 = 4\), the general slope \(\frac{dy}{dx}\) is \(\frac{2x - y}{x - 2y}\).
To find the slope of the tangent line at a specific point \((a, b)\), simply substitute \(a\) and \(b\) into the slope formula. Equivalently, for the point \((-a, -b)\), substitute these values to find that both slopes are indeed the same. Therefore, the tangents at \((a, b)\) and \((-a, -b)\) are parallel.

Understanding Conic Sections

Conic sections are the curves obtained by intersecting a plane with a double-napped cone. Types of conic sections include ellipses, hyperbolas, parabolas, and circles, each with unique properties.
An ellipse, such as the one defined by the equation \(x^2 - xy + y^2 = 4\), is one example. It typically features two axes of symmetry—major and minor axes—but in the presented scenario, its symmetry extends to the origin.

• Ellipses bound closed curves and have two foci.
• The sum of the distances from any point on the ellipse to the foci is constant.
• The general form of an ellipse's equation is \(ax^2 + bxy + cy^2 = d\) where \(b^2 < 4ac\).

An ellipse in analytical geometry appears as a structurally balanced, symmetrical shape, making it a fascinating and intricate conic section to work with.