Problem 71 - Contemporary Precalculus

Question

If \(P\) is a point on a circle with center \(C\), then the tangent line to the circle at \(P\) is the straight line through \(P\) that is perpendicular to the radius \(C P\). In Exercises \(67-70\), find the equation of the tangent line to the circle at the given point.)Let \(A, B, C, D\) be nonzero real numbers. Show that the lines \(A x+B y+C=0\) and \(A x+B y+D=0\) are parallel.

Step-by-Step Solution

Answer

Question: Prove that the given lines \(Ax+By+C=0\) and \(Ax+By+D=0\) are parallel. Answer: We determined the slopes of both lines to be \(-\frac{A}{B}\). Since their slopes are the same, the lines are parallel.

Step 1: Determine the slope of the first line

To find the slope of the first line, we can rewrite the equation in the form \(y=-\frac{A}{B}x-\frac{C}{B}\), which is a form of a linear equation. The slope of the first line is \(-\frac{A}{B}\).

Step 2: Determine the slope of the second line

Similar to the first line, we can rewrite the second line's equation in the form \(y=-\frac{A}{B}x-\frac{D+C}{B}\). The slope of the second line is also \(-\frac{A}{B}\).

Step 3: Compare the slopes

The slopes of both lines are \(-\frac{A}{B}\). Since the slopes are the same, the given lines are parallel.

Key Concepts

Equation of a Tangent Line

The equation of a tangent line to a circle is a fascinating concept in geometry. Imagine a circle with a center at point \(C\) and a point \(P\) on the circle's circumference. A tangent line is a straight line that just "touches" the circle at this point \(P\).
Here's something special about tangent lines: at the point of contact \(P\), they are perpendicular to the radius \(CP\). This unique relationship helps us establish the equation for the tangent line.
To find this equation, you need a point-slope form which is based on:

• The point \(P(x_1, y_1)\) on the circle
• The slope of the tangent line

Since the tangent line is perpendicular to the radius, if you know the slope of the radius, the tangent's slope will be the negative reciprocal of that. If the radius has a slope \(m\), the tangent line's slope will be \(-\frac{1}{m}\). This approach simplifies establishing the tangent line's equation.
Understanding this connection between the radius and the tangent line is key to mastering this concept.

Parallel Lines

Parallel lines are an intriguing aspect of geometry where two lines run side by side and never meet. The main trait of parallel lines is that they have the same slope.
When you look at the equations \(Ax + By + C = 0\) and \(Ax + By + D = 0\), they represent two lines. To grasp why they are parallel, convert each into the slope-intercept form \(y = mx + b\). You'll see both lines have the same slope \(-\frac{A}{B}\).
This discovery confirms they are parallel because:

• Same slope means the rise over run is identical
• Parallel lines indicate a consistent distance apart

So even though \(C\) and \(D\) impact the position of each line, they don't alter the slope. That’s why these lines stay parallel, ensuring they won't intersect no matter how far they're extended.

Slope of a Line

The slope of a line is fundamental in understanding linear equations and graphs. It measures how steep a line is and is often expressed as the ratio of vertical change (rise) to horizontal change (run).
For any linear equation in the form \(y = mx + b\), the slope \(m\) tells us a lot:

• A positive slope means the line ascends from left to right
• A negative slope means the line descends from left to right
• A zero slope results in a horizontal line
• An undefined slope is associated with a vertical line

In the context of comparing lines, knowing the slope helps identify properties like parallelism or perpendicularity. For example, if two lines have the same slope, they’re parallel. And if one's slope is the negative reciprocal of another's, they’re perpendicular.
This concept of slope, when embraced, becomes a powerful tool in analyzing and predicting the behavior of linear equations on a graph.