Problem 63 - Precalculus

Question

The parabolic cross section of a satellite dish can be modeled by a portion of the graph of the equation $$x^{2}-2 x y-27 \sqrt{2} x+y^{2}+9 \sqrt{2} y+378=0$$ where all measurements are in feet. (a) Rotate the axes to eliminate the \(x y\) -term in the equation. Then write the equation in standard form. (b) A receiver is located at the focus of the cross section. Find the distance from the vertex of the cross section to the receiver.

Step-by-Step Solution

Answer

The distance from the vertex of the parabolic cross section to the receiver is 81 feet.

Step 1: Rotating the axes

Since coefficients of \(x\) and \(y\) are equal in the equation i.e. \(-27\sqrt{2}\), the angle of rotation \(\theta\) would be \(\frac { \pi } { 4 }\). We can eliminate the \(xy\) term in the equation by using the rotation formulas \(x = Xcos\theta - Ysin\theta\) and \(y = Xsin\theta + Ycos\theta\) where \(X\) and \(Y\) are new coordinates, to get new equation in terms of \(X\) and \(Y\).

Step 2: Writing the equation in standard form

After replacing \(x\) and \(y\) in terms of \(X\) and \(Y\) in the equation and simplifying using the decimals for either of the angles \(\theta = 135^{°}\) or \(\theta = 315^{°}\), the equation should be simplified to \(Y^{2} + 324 = 2X\) which simplifies into \[(Y - 0)^{2} = 4*81*(X - 0) \] or \[ (X - 0)^{2} = 4*81*(Y - 0)\] This is the standard form of the equation.

Step 3: Finding focurs

The above equation is of the form \( (Y - k)^{2} = 4a(X - h) \) or \( (X - h)^{2} = 4a(Y - k)\) where \(h = 0\), \(k = 0\), so the vertex of the parabola is at origin (0, 0). The value of \(a\) is half of the coefficient of \(X\) or \(Y\) in the equation and is equal to 81.

Step 4: Finding the distance of receiver from the vertex

As the vertex of the parabola is at origin (0, 0) and \( a = 81 \), the coordinates of the focus would be \((a, 0) = (81, 0)\), if the parabola opens to the right or \((0, a) = (0, 81)\), if the parabola opens upwards. Hence, the receiver is located 81 feet from the vertex in either case.

Key Concepts

Axis Rotation

When trying to simplify the equation of a parabola that includes an \(xy\) term, we can make the curve easier to analyze by eliminating this term. This process is called axis rotation. By rotating the coordinate axes, we transform the graph. For the given equation, where both the coefficients of the \(x\) and \(y\) terms are the same, we use a rotation angle \(\theta\) of \(\frac{\pi}{4}\) or 45 degrees. To perform axis rotation:

• Replace \(x\) and \(y\) with their rotated equations: \(x = X \cos \theta - Y \sin \theta\) and \(y = X \sin \theta + Y \cos \theta\).
• Substitute these into the original equation to get a new equation in terms of \(X\) and \(Y\).

By converting the equation in this manner, you essentially align the parabola with one of the principal axes, which simplifies further analysis.

Standard Form Equation

After the axes have been rotated and the equation has been transformed into \(X\) and \(Y\) coordinates, the goal is to re-write it in the standard form. The standard form for a parabola generally follows: For parabolas opening sideways: \((Y - k)^2 = 4a(X - h)\)For parabolas opening upwards or downwards:\((X - h)^2 = 4a(Y - k)\)In the given solution, the equation becomes \(()^2 = 4 \times 81 (X - 0)\) or \((X - 0)^2 = 4 \times 81 (Y - 0)\).The numbers \(h\) and \(k\) denote the coordinates of the vertex of the parabola and \(a\) is a measure which indicates the distance from the vertex to the focus. Writing equations in this manner helps us quickly identify these key properties of a parabola.

Vertex of a Parabola

The vertex is an important element of a parabola as it represents either the highest or lowest point depending on its orientation. For standard forms:

• If a parabola opens sideways, as in \((Y - k)^2 = 4a(X - h)\), the vertex is \((h, k)\).
• If it opens upwards or downwards, \((X - h)^2 = 4a(Y - k)\), the vertex is \((h, k)\).

In this exercise, after formulating the equation, we find that the vertex is at the origin \((0,0)\). This makes calculating other properties, like the distance to the focus, straightforward since all measurements can hinge directly from the zero point.

Focus of a Parabola

The focus of a parabola is another critical point that lies along the axis of symmetry, around which the parabola is shaped. This is where reflected signals, like radio waves from a satellite dish, converge.For the equation forms:

• In \((Y - k)^2 = 4a(X - h)\), the focus is \((h+a, k)\) if it opens right, or \((h-a, k)\) if it opens left.
• In \((X - h)^2 = 4a(Y - k)\), the focus is \((h, k+a)\) if it opens upwards, or \((h, k-a)\) if it opens downwards.

Given the solution, with the vertex at \((0,0)\) and \(a = 81\), the focus is either \((81, 0)\) or \((0, 81)\). This means the receiver, or focus in this scenario, is 81 feet from the vertex, confirming the parabola's characteristic as an effective device for signal collection.