Problem 54 - Calculus Single Variable

Question

Determine a value of \(k\) for which the graph of the given Cartesian equation is a point. \(x^{2}+y^{2}-16 x+k=0\)

Step-by-Step Solution

Answer

The value of \(k\) for which the graph is a point is \(k = 64\).

Step 1: Identify the Condition

For the graph of the equation to be a point, the equation must represent a single point, which occurs when the radius is zero. This requires completing the square for the equation.

Step 2: Rearrange the Equation

Start with the given equation: \(x^{2}+y^{2}-16x+k=0\). Collect the constant terms on one side: \(x^{2} - 16x + y^{2} = -k\).

Step 3: Complete the Square for x

Consider the terms involving \(x\): \(x^{2} - 16x\). To complete the square, add and subtract \(8^2 = 64\), resulting in: \( (x - 8)^2 - 64 \).

Step 4: Express the Equation in Standard Form

Substitute the completed square back into the equation: \((x - 8)^2 - 64 + y^2 = -k\). Simplify to get: \((x - 8)^2 + y^2 = 64 - k\).

Step 5: Set the Radius to Zero

A point has a radius of zero. Therefore, set: \((x - 8)^2 + y^2 = 0\), which implies \(64 - k = 0\).

Step 6: Solve for k

Solve \(64 - k = 0\) to find \(k\). You get \(k = 64\).

Key Concepts

Completing the Square

Completing the square is a handy technique used in algebra to transform a quadratic expression into a perfect square trinomial. This is significant when dealing with quadratic equations like those representing circles in the Cartesian coordinate system.

To complete the square, we follow these steps:

• Identify the quadratic expression in the form of \(x^2 + bx\).
• Take half of the coefficient of \(x\), square it, and add and subtract this square to the expression. This essentially means you have formed \((x - h)^2\) where \(h\) is half of the \(b\) coefficient.
• In our example, for the expression \(x^2 - 16x\), we have \(b = -16\). Taking half gives us \(-8\) and squaring it gives \(64\). So, we add and subtract \(64\) to complete the square, resulting in \((x - 8)^2 - 64\).

This process transforms the equation into a format that makes it easier to interpret, especially when analyzing the geometric properties of circles in the coordinate plane.

Radius

In the context of a circle, the radius is the distance from the center of the circle to any point on its circumference. In the Cartesian coordinate system, a circle's equation is typically written in the form \((x - h)^2 + (y - k)^2 = r^2\) where \((h, k)\) is the center of the circle and \(r\) is the radius.

To find whether the equation represents a circle and what its radius is:

• Ensure the equation is in the standard form of a circle.
• The expression \((x - h)^2 + (y - k)^2\) is set equal to \(r^2\), the square of the radius.
• From the completed formation \((x - 8)^2 + y^2 = 64 - k\), it's clear that the expression represents a circle because of the sum of squares form.

Setting the radius, \(r^2\), to zero is what signifies that there is no distance from a center, thus making the graph an individual point instead of a circle. For our exercise, when we solved \(64 - k = 0\), we found that \(k = 64\) ensured the radius was zero.

Graph of an Equation

Graphs of equations are visual representations in a coordinate system that illustrate all solutions of an equation. For example, a circle's equation graphically represents all the points equidistant from a center point. Different equations produce different graphs such as lines, parabolas, and circles.

In our specific exercise:

• The initial equation is \(x^{2}+y^{2}-16x+k=0\), initially it's not clear what form it takes.
• Through completing the square, the equation was restructured to \((x - 8)^2 + y^2 = 64 - k\), indicating a potential circle graph.
• However, by determining the condition \(64 - k = 0\), we established the graph was not truly a circle but a singular point when \(k=64\).

A single point on a graph suggests a radius of zero, deviating from a regular circular graph symmetry. Understanding the graph helps visualize solutions or lack thereof, quickly revealing characteristics such as intercepts, symmetry or behavior patterns.