Finding the Set of Values for Which a Line Does Not Intersect a Curve

Finding the Set of Values for Which a Line Does Not Intersect a Curve
Finding the Set of Values for Which a Line Does Not Intersect a Curve

If you have ever worked with linear or non-linear equations, you know that lines and curves can intersect in many ways. However, in some cases, you may want to find a set of values for which a line does not intersect a curve at all.

Finding the set of values for which a line does not intersect a curve is a common question when taking A level maths. It can be important in fields including engineering, physics, and finance, where you may need to calculate boundaries or constraints to optimize your project.

Read on to find out how to find the set of values for which a line does not intersect a curve. You might also enjoy reading: 9 Best YouTube Channels to Learn Calculus.

Differences Between a Line And a Curve

The main difference between a line and a curve is that a line is generally straight, whereas a curve is considered a generalization of a line.

Generally, a line can be defined by its slope-intercept formula y = mx + b, where m is the slope and b is the y-intercept. A curve, on the other hand, can have multiple forms depending on its degree, but for simplicity, we will focus on a quadratic curve y = ax^2 + bx + c.

How Do You Show That a Line Does Not Intersect a Curve?

A line does not intersect a curve if the discriminant of the resulting quadratic equation is negative. To know if a line intersects a curve, we need to find the values of x and y that satisfy both equations simultaneously. This means we need to solve the following system of equations:

y = mx + b, y = ax^2 + bx + c

To find a set of values for which a line does not intersect a curve, we need to manipulate one of the equations to obtain a condition that excludes the intersection. For instance, in the case of a quadratic curve, we can use the discriminant Δ = b^2 – 4ac to determine the number of roots (x values) that the equation can produce.

  • If Δ is positive, the curve has two distinct roots, which means the line can intersect the curve twice.
  • If Δ is zero, the curve has one repeated root, which means the line intersects the curve once. In other words, the line is tangent to the curve.
  • If Δ is negative, the curve has no real roots, which means the line does not intersect the curve at all. Therefore, the condition we need to satisfy is Δ < 0.

Examples Of Finding the Set of Values for Which a Line Does Not Intersect a Curve

Whether you’re a math novice or a seasoned expert, finding the set of values for which a line does not intersect a curve can be a difficult task. I believe that understanding how lines and curves interact can help us solve complex problems in a variety of fields, from engineering and physics to finance and statistics.

Here are the steps you can take to find the set of values in which they do not intersect, along with the mathematical concepts behind them and a few practical examples to help you understand the subject thoroughly.

1- Using the Quadratic Method

Suppose we have the line y = 3x – 2 and the curve y = x^2 + 4. We want to find the set of values for which the line does not intersect the curve. We can start by substituting the line equation into the curve equation:

3x – 2 = x^2 + 4

Rearranging, we get:

x^2 – 3x + 6 = 0

We can now calculate the discriminant Δ:

Δ = (-3)^2 – 4(1)(6) = -15

Since Δ is negative, there are no real roots, which means the line does not intersect the curve at all. Therefore, the set of values for which the line does not intersect the curve is the entire real line.

2- Using Geometry

Another way to obtain a condition that excludes the intersection is to use the geometry of the problem. Consider the case where the curve is a circle centered at the point (a, b) with radius r. If the line is perpendicular to the radius passing through the center, it will not intersect the circle.

To see why, imagine the circle as a clock face and the line as the hour hand. The hour hand can sweep the whole clock face without intersecting the center if it points to the opposite side of the circle. This means that the slope of the line is the negative reciprocal of the slope of the radius passing through the center, which is (y – b)/(x – a). Therefore, the condition we need to satisfy is:

m = -(x – a)/(y – b)

Let’s illustrate this concept with an example. Suppose we have the line y = -2x + 5 and the circle (x – 2)^2 + (y – 3)^2 = 9. We want to find the set of values for which the line does not intersect the circle. We can start by rearranging the line to the slope-intercept form:

y = -2x + 5

The slope of the line is -2, which means the slope of the radius passing through the center (2, 3) is 1/2. Therefore, the negative reciprocal slope of the line is 2. We can now substitute the coordinates of the center and the slope of the radius into the equation to obtain the following:

2 = -(x – 2)/(y – 3)

Rearranging, we get:

2(y – 3) + x – 2 = 0

We can now compare this equation to the general form of a circle:

(x – a)^2 + (y – b)^2 = r^2

We see that the coefficients of x, y, and the constant term are a = 2, b = 3, and r^2 = 10. Therefore, the set of values for which the line does not intersect the circle is:

(x – 2)^2 + (y – 3)^2 > 10

3- Using Calculus

Another way to determine where a line does not intersect a curve is to use calculus. By taking the derivative of both the line and the curve, we can determine the slope of each equation.

  • If the slopes of the line and the curve are equal, which means they are parallel; therefore, they do not intersect.
  • If the product of the two slopes is – 1, it means they are perpendicular and intersecting. This method is more accurate and can be used for more complex curves.

To learn more, I encourage you to watch the video below.

What to read next:

Wrapping Up

Finding the set of values for which a line does not intersect a curve requires manipulating the equations to obtain a condition that excludes the intersection.

I discussed finding the set of values for which a line does not intersect a curve using the discriminant of a quadratic equation, the geometry of a circle, and calculus.

I believe that by following the steps discussed in this article, you can solve complex problems and better understand how lines and curves interact in everything from physics to finance, including optimization within certain constraints.

Altiné

I am Altiné. I am the guy behind mathodics.com. When I am not teaching math, you can find me reading, running, biking, or doing anything that allows me to enjoy nature's beauty. I hope you find what you are looking for while visiting mathodics.com.

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