Problem 77 - College Algebra

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

Step 1: Determine the Slope of Radius

The radius of the circle at the point of tangency a straight line extending from the origin to the point \((3, -4)\). This line has a slope of \(-4/3\)

Step 2: Find the Slope of the Tangent Line

Since the tangent line is perpendicular to the radius at the point of tangency, the slope of the tangent line is the negative reciprocal of the slope of the radius. Therefore, the slope \(m\) of the tangent line is \(3/4\).

Step 3: Write the Equation of the Tangent Line

The equation of a line in point-slope form is \(y - y_{1} = m(x - x_{1})\). Substituting the slope \(m = 3/4\), and the point \((3, -4)\) into this equation, we get \(y + 4 = 3/4 (x - 3)\).

Key Concepts

Point-Slope Form

The point-slope form is a handy way to write the equation of a line. It's especially useful when you know a point on the line and its slope. The formula for the point-slope form is: \( y - y_1 = m(x - x_1) \) where:

• \( m \) is the slope of the line
• \( (x_1, y_1) \) is a point on the line

Using this form, you can quickly write the line’s equation without needing to know its y-intercept. In our exercise, once we calculated the slope of the tangent line to be \( \frac{3}{4} \), we used the point \((3, -4)\) to form the equation: \( y + 4 = \frac{3}{4}(x - 3) \). It’s simple: plug in the slope and the point into the formula, and there you have it!
For those moments when you have a point and a slope, but need the entire line equation, point-slope form is your best friend.

Circle Equation

The equation of a circle is a mathematical way to describe all the points that create a perfect circle. The standard form is: \( x^2 + y^2 = r^2 \) where \( r \) is the radius of the circle.
In our problem, we had a circle defined by \( x^2 + y^2 = 25 \). Here, the radius squared \( r^2 \) is \( 25 \), indicating a circle with a radius of \( 5 \). This simple equation beautifully captures the symmetry and balance of a circle centered at the origin \((0,0)\).
While it might look neat on paper, this form is powerful in connecting a multitude of mathematical ideas and offers a gateway to understanding circular motion, geometry, and conics.

Slope of a Line

The slope of a line measures its steepness. It's determined by the ratio of the vertical change (rise) to the horizontal change (run), often written as \( \frac{\Delta y}{\Delta x} \). When you have two points \((x_1, y_1)\) and \((x_2, y_2)\), the slope \( m \) is calculated like this: \( m = \frac{y_2 - y_1}{x_2 - x_1} \)
In our scenario, the concept of slope helps us find both the slope of the radius and the tangent line. For the radius from the origin to point \((3, -4)\), we get a slope of \(-\frac{4}{3}\). This tells us that for every 3 units we move right, we move 4 units down.
Stable and reliable, the slope provides a numeric understanding of angle and direction in a plane, essential for algebra and calculus alike.

Perpendicular Lines

Perpendicular lines intersect to form a right angle. A key feature of perpendicular lines in geometry is that their slopes are negative reciprocals. If one line has a slope \( m \), then a line perpendicular to it will have a slope of \( -\frac{1}{m} \).
In our exercise, we found that the slope of the radius was \(-\frac{4}{3}\). Therefore, the tangent line, being perpendicular, has a slope of \( \frac{3}{4} \). This relationship allows us to solve problems involving angles and intersections of lines on a plane.
Recognizing when two lines are perpendicular is crucial for understanding geometric properties and relationships, bridging the gap between algebraic equations and geometric intuition.