Problem 45 - Calculus: One and Several Variables

Question

Determine the slope of the line that intersects the circle \(x^{2}+y^{2}=169\) only at the point (5,12)

Step-by-Step Solution

Answer

The slope of the line that is tangent to the circle \(x^{2}+y^{2}=169\) at the point (5,12) is -5/12.

Step 1: Determine the slope perpendicular to the desired line

The line connecting the center of the circle and the point of tangency is a radius. Compute the slope of this line utilizing the formula \(m=(y_{2}-y_{1})/(x_{2}-x_{1})\). Using the points (0,0) and (5,12), the formula becomes \(m=(12-0)/(5-0)\), so \(m=12/5\).

Step 2: Determine the slope of the desired line

The slope of the line tangent to the circle is the negative reciprocal of 12/5, which is -5/12. This is because the slopes of perpendicular lines are negative reciprocals of each other.

Key Concepts

Slope of a Line

The slope of a line is a measure of its steepness, therefore it's crucial for understanding many geometrical concepts. When dealing with a line on a coordinate plane, the slope is calculated using two points on the line. The formula for the slope, denoted as \(m\), is:

• \(m = \frac{y_2 - y_1}{x_2 - x_1}\)

This formula helps you compare the vertical change (rise) to the horizontal change (run) between two points \((x_1, y_1)\) and \((x_2, y_2)\). A positive slope means the line is rising, while a negative slope means it is falling.

Understanding slope is fundamental for analyzing how a tangent line behaves compared to a radius or any other line it interacts with.

Circle Equation

A circle on a coordinate plane can be described using the circle equation \(x^2 + y^2 = r^2\), where \((x, y)\) are the coordinates of points on the circle, and \(r\) is the radius. The center of this particular circle is at \((0, 0)\) and the radius is \(13\), since \(r^2 = 169\) equals \(13^2\).

This equation is essential for identifying any point lying on the circle's circumference. If you have a specific point, such as \((5,12)\), you can verify it lies on the circle by substituting it into the equation, ensuring the equation holds true.

Understanding the circle equation is crucial for problems involving tangent lines, as it helps determine the relationship between the radius, tangent, and points on the circle.

Negative Reciprocal

The concept of a negative reciprocal is vital when dealing with perpendicular lines in geometry. The negative reciprocal of a slope is determined by flipping the fraction's numerator and denominator and changing the sign. For example, if the slope \(m\) is \(\frac{a}{b}\), its negative reciprocal will be \(-\frac{b}{a}\).

This concept is used to find the slope of a tangent line to a circle. If the slope of the radius (the line from the center of the circle to the point of tangency) is known, the tangent line's slope can be easily found.

• Radius slope: \(\frac{12}{5}\)
• Tangent slope: \(-\frac{5}{12}\)

Understanding the negative reciprocal allows us to move between finding slopes of intersecting lines and is especially useful for tangent calculations.

Tangent and Radius Relationship

The relationship between a tangent line and a radius of a circle is key in many geometric concepts. A tangent line to a circle is one that touches the circle at exactly one point. At the point of tangency, the tangent line is perpendicular to the radius derived from the center of the circle to this point.

In this context, to solve our problem, we first establish the slope of the radius (from the center \((0,0)\) to the tangency point \((5,12)\)). Its slope is \(\frac{12}{5}\). Since a tangent is perpendicular to the radius, the slope of the tangent is the negative reciprocal, which is \(-\frac{5}{12}\).

Understanding this relationship invites a deeper appreciation of how shapes and lines behave on the coordinate plane and is central to mastering the topic of tangents to circles.