Problem 97 - Precalculus Essentials

Question

A tangent line to a circle is a line that intersects the circle at exactly one point. The tangent line is perpendicular to the radius of the circle at this point of contact. Write an equation in point-slope form for the line tangent to the circle whose equation is \(x^{2}+y^{2}=25\) at the point (3,-4).

Step-by-Step Solution

Answer

The equation of the tangent line to the circle \(x^{2}+y^{2}=25\) at the point (3,-4) is \(y + 4 = 3/4 (x - 3).\

Step 1: Find the slope of the radius

The radius here connects the origin (0, 0) and the point (3, -4). Its slope (m) is determined by the formula \(m = (y_2 - y_1)/(x_2 - x_1) = (-4 - 0)/(3 - 0) = -4/3.\)

Step 2: Find the slope of the tangent

As the tangent line is perpendicular to the radius at the point of contact, the slope of the tangent line is the negative reciprocal of the slope of the radius, i.e., 3/4.

Step 3: Write the equation of the tangent in point-slope form

The point-slope form of a line is \(y - y_{1} = m(x - x_{1})\), where m is the slope and (x1, y1) is a point on the line. Using the slope of the tangent line calculated in Step 2, and the point of contact (3,-4), the equation of the tangent line is \[y - (-4) = 3/4 (x - 3)\] or \[y + 4 = 3/4 (x - 3).\]\

Key Concepts

Understanding Point-Slope Form

The point-slope form of a linear equation is very handy when you know a point on a line and the line's slope. It's written as \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a known point through which the line passes.
This form is particularly useful for writing equations of tangent lines to circles because once the slope and point of contact are known, we can plug them directly into the formula.
For example, if you need to find the equation of a line tangent to a circle at a specific point, like in our problem, the point-slope form provides a straightforward method to construct this equation using the given point and the slope of the tangent.
It's a universal tool that simplifies many problems involving linear equations.

Perpendicular Lines and Their Slopes

Perpendicular lines intersect at right angles, which is 90 degrees. In terms of slopes, if two lines are perpendicular, the product of their slopes is always -1. This gives us an easy way to calculate the slope of one line if we know the slope of a line perpendicular to it.
For instance, if you have a line with a slope of \( -4/3 \), as from a radius of a circle, the slope of a line perpendicular to it (like a tangent) would be the negative reciprocal. This is because \((-4/3) \times m = -1\) simplifies to \( m = 3/4 \).
Understanding this relationship is crucial in problems involving circles, particularly when constructing tangent lines as it allows us to easily find the correct slope.

What is a Negative Reciprocal?

The negative reciprocal of a number is found by first flipping the numerator and denominator of a fraction and then changing the sign. This concept is central in determining the slope of lines perpendicular to each other.
For example, the negative reciprocal of \(-4/3\) is \(3/4\).
By calculating a negative reciprocal, you are essentially finding the slope of a line that would cross another at a 90-degree angle. This is a key concept for determining the direction of lines, such as perpendicular tangents from radii to circles.
Understanding negative reciprocals helps in transforming one line's slope into that of another line at a right angle, simplifying problems like those involving tangents to a circle.

Equations of a Circle

The equation of a circle in standard form is \(x^2 + y^2 = r^2\), where \(r\) is the radius. This equation describes a circle centered at the origin where the distance from any point \((x, y)\) on the circle to the center is equal to \(r\).
In our problem, the circle equation is \(x^2 + y^2 = 25\), meaning our circle has a radius of 5 since \(r^2 = 25\).
Knowing the circle equation is essential when dealing with tangents, because it helps in determining other necessary elements, like the radius slope, to find the tangent line equation.
Equations of circles also help identify key characteristics of a circle, such as its size and position, making them a fundamental part of geometry.