While preparing for the ACT, you might encounter problems that mention the words “conics” or “conic sections”. If you’re looking for more information on how to ace answering questions involving parabolas, hyperbolas, ellipses, and circles, you’ve come to the right place. In this article, we’ll summarize all the things you need to know about conic sections. Get ready to master this topic for your upcoming ACT!
What Is a Conic Section? Everything You Need To Know About Conic Sections!
Conic sections are the different possible sections formed when a cone is sliced by a plane at different angles. For the ACT, there are four conic section formulas and graphs that we’ll be focusing on circles, ellipses, parabolas, and hyperbolas. Take a look at the diagram below to see how each conic section was formed.
In this article, let’s explore what makes each of the four conic sections unique. Take note of what makes each of the four conic section formulas unique and learn each of their key components. By the end of the article, feel confident to take on ACT math questions involving the four types of conic sections.
What Are the Four Types of Conic Sections in the ACT?
You’ll need to know four important types of conic sections for the ACT: circle, parabola, ellipse, and hyperbola. For each conic section, we’ll lay out the standard form, its graph, and the components that you need to know to graph the conic section on your own.
You might have encountered circles and the equations of circles earlier in your ACT math prep. Yes, it’s a special type of conic section because it’s symmetric across its shape. From the diagram we’ve shown earlier, the circle is the result when the plane intersects with a cone parallel to the base.
One of the key features of circles is the fact that the center, (h, k), is equidistant from any point on the circle. The segment that connects between the center and the point is the circle’s radius, r. The equation for the circle is equal to (x – h)2 + (y – k)2 = r2, where (h, k) is the circle’s center and r represents the circle’s radius.
The parabola that you’ve encountered in the past is the graph formed by a quadratic equation. This, is in fact, a conic section formed when the plane cuts the cone at an inclined angle as shown in the earlier diagram. The standard parabola opens upward vertically and when the leading coefficient is negative, it opens downward vertically as shown below.
Given the vertex of the parabola, (h, k), and the leading coefficient of the function, a, its equation is equal to y = a(x – h)2 + k. As mentioned, when a > 0, the parabola opens upward and when a < 0, the parabola opens downward.
The ellipse looks like a circle, but unlike circles, it is only symmetric about the x and y axes. It is formed when the plane cuts the cone at an angle and forms an oval-shaped figure as shown in the first diagram. What makes ellipses special are their foci (or focus in singular terms) – this determines whether the ellipsis is longer vertically or horizontally.
These are two possible orientations of the ellipse: when stretches vertically and when it stretches horizontally. This is based on the denominators found in the fractions containing x2 and y2.
- When the denominator below x2 or (x-h)2 is bigger, assign a to be the denominator. This represents the longest distance of a point on the ellipse from the center, (h, k). The major axis is then equal to 2a. This means that the minor axis is 2b long and the vertical distance of the point from the center is b. Hence, the equation of the ellipse is (x – h)2a2 + (y – k)2b2=1.
- Swap these components when it’s the other way around: y2 or (y-k)2 contains the larger denominator. This means that the ellipse stretches vertically with 2a as the length of the major axis and 2b as the length of the minor axis. The only difference is that the major axis is now parallel to the y-axis. The equation of the ellipse is now (x – h)2b2 + (y – k)2a2=1.
The hyperbola’s general equation is similar to the ellipse. The only difference is that it has a minus sign in the middle. The hyperbola’s graph is similar to that of the parabola, but this time, it contains two parabolas that either open vertically or horizontally. How does this happen? The hyperbola is formed when the plane intersects with the cone parallel to its axes. Hence, we have the two standard forms for the hyperbolas as shown below.
Like parabolas, each of the U-shaped curves has a vertex defined by the distances, a and b, from the center (h, k). If x2 or (x -h)2 is the leading term and with a positive sign, the hyperbola opens sidewards with the centers a units away from (h, k). Meanwhile, when y2 or (y -k)2 is the leading expression in the standard form, the hyperbolas open upward and downward. Each of the centers is b units away from (h, k). Hence, we have the following equations for the hyperbolas:
- (x – h)2a2 – (y – k)2b2=1is a hyperbola that opens sidewards with each curve a units away from the center, (h, k).
- (y – k)2b2 – (x – h)2a2=1is a hyperbola that opens upward and downward with each curve b units away from the center, (h, k).
There are still a lot of components and parameters that make each of these four conic sections unique. These are the fundamental things you need to know about the circle, parabola, ellipse, and hyperbola. In the next section, it’s time for you to see how to use these properties of conic sections to solve some ACT math questions.
How To Use the Conic Section Formulas To Solve Some ACT Problems?
When given an ACT math problem involving conic section equations, the first step is to identify which conic section we’re working with. Then, use what you’ve just learned about each of the four conic sections to answer the given question. Identify key concepts from the given question then apply the properties correctly.
ACT Question 1: Finding the Equation of the Conic Section Given Its Graph
There are instances when you’re asked to establish the equation of the conic section given its graph. Take a look at the question below and try it out for yourself!
In the standard (x, y) coordinate plane, the center of the circle lies on the y-axis as shown below. If the circle is tangent to the x-axis, which of the following is the equation of the circle?
A. x2+ y2=4
B. (x-2)2+ y2=1
C. x2+ (y-2)2=1
D. x2+ (y-2)2=4
E. (x-2)2+ y2=4
From the graph of the circle, we can see that the center lies on (0, 2). The distance of the center from the endpoint of the circle is 2 units. This means (h, k) = (0, 2) and r = 2, so use these values in the standard form of the circle’s equation, (x-h)2+(y-k)2=r2. Hence, the equation of the circle is x2+(y-2)2=4. This makes D the correct answer.
ACT Question 2: Solving a Problem That Requires Us to Graph the Conic Section
Let’s take the level up a notch! Try solving this problem where you’re expected to know how a circle and ellipse are graphed on an xy-plane.
Suppose the equations (x-2)2+(y-3)2=1 and (x – 3)24 + (y – 2)29=1 are graphed in the same standard (x, y)-plane. How many points of intersections do these graphs share?
While it’s possible to find the points of intersection by solving the equations simultaneously, that will be tedious. This is why knowing conic sections equations and their graphs helps us solve trick ACT questions like this! First, let’s identify the conic sections that these equations represent:
- (x-2)2+(y-3)2=1 is a circle with a center at (2,3 ) and a radius of 1.
- (x – 3)24 + (y – 2)29=1 is an ellipse centered at (3,2). Since (y-2)2 has the bigger denominator, the ellipse is stretched vertically with a major axis that is 23=6 units long.
Graph these two conic sections in one plane and see if they intersect at certain points. The graph below confirms that the two equations intersect at exactly two points.
This shows that the correct answer is C. By applying what you know of conic sections, you get to solve problems faster! Enjoying these practice questions? Don’t worry, AceIt offers unlimited practice questions on different math topics including the conics section to help you ace your ACT. Sign up for a free trial to see for yourself!
Acing the Conics Section for Your ACT Success
We’ve now established the fundamentals of conics sections to help you ace the ACT’s math section. By knowing the graphs and equations of the different conic sections, you can solve different types of questions easily. Bookmark this article in case you want to review key concepts about circles, parabolas, ellipses, and hyperbolas. Don’t forget to try working on the sample problems we’ve prepared for you as well. Embrace the challenge of mastering and acing the conics sections to pave the way towards your ACT success. You got this!