A parabola is a plane curve formed by a point moving so that its distance from a fixed point is equal to its distance from a fixed-line. The fixed-line is the directrix of the parabola and the fixed point is the focus denoted by F. The axis of the parabola is the line through the F and perpendicular to the directrix. The point where the parabola intersects the axis is called the vertex of the parabola. In this article, we will learn how to find the vertex focus and directrix of the parabola with the given equation. Show
Step 1. Determine the horizontal or vertical axis of symmetry. Step 2. Write the standard equation. Step 3. Compare the given equation with the standard equation and find the value of a. Step 4. Find the focus, vertex and directrix using the equations given in the following table. The following table gives the equation for vertex, focus and directrix of the parabola with the given equation.
Video LessonVertex and Directrix of Parabola Solved ExamplesLet us have a look at some examples. Example 1: Find the vertex, focus, the equation of directrix and length of the latus rectum of the parabola y2 = -12x. Solution: Given equation of parabola is y2 = -12x …(i) This equation has y2 term. So the axis of the parabola is the x-axis. Comparing (i) with the equation y2 = -4ax We can write -12x = -4ax So a = 12/4 = 3 Focus is (-a,0) = (-3,0). Equation of directrix is x = a. I.e x = 3 is the required equation for directrix. Vertex is (0,0). Length of latus rectum = 4a = 4×3 = 12. Example 2. Given the equation of a parabola 5y2 = 16x, find the vertex, focus and directrix. Solution: Given equation is 5y2 = 16x y2 = (16/5)x Comparing above equation with y2 = 4ax We get, 4a = 16/5 a = ⅘. Focus is (a,0) = (4/5,0). Equation of directrix is x = -a. I.e x = -⅘ Vertex is (0,0). Example 3. How to find the directrix, focus and vertex of a parabola y = ½ x2. Solution: Given equation is y = ½ x2 Rearranging we get x2 = 2y The axis of the parabola is y-axis. Comparing the given equation with x2 = 4ay We get 4ay = 2y a = 2/4 = ½ Focus is (0,a) = (0, ½ ) Equation of directrix is y = -a. i.e. y = -½ is the equation of directrix. Vertex of the parabola is (0,0). Related Links:
Frequently Asked QuestionsParabola is a locus of a point, which moves so that distance from a fixed point (focus) is equal to the distance from a fixed line (directrix). Vertex is the point where the parabola intersects the axis. The equation of the directrix of the parabola y2=4ax is given by x = -a. The focus of the Parabola y2=4ax is (a, 0). In mathematics, a parabola is the locus of a point that moves in a plane where its distance from a fixed point known as the focus is always equal to the distance from a fixed straight line known as directrix in the same plane. Or in other words, a parabola is a plane curve that is almost in U shape where every point is equidistance from a fixed point known as focus and the straight line known as directrix. Parabola has only one focus and the focus never lies on the directrix. As shown in the below diagram, where P1M = P1S, P2M = P2S, P3M = P3S, and P4M = P4S. Equation of the parabola from focus & directrixNow we will learn how to find the equation of the parabola from focus & directrix. So, let S be the focus, and the line ZZ’ be the directrix. Draw SK perpendicular from S on the directrix and bisect SK at V. Then, VS = VK The distance of V from the focus = Distance of V from the directrix V lies on the parabola, So, SK = 2a. Then, VS = VK = a Let’s take V as vertex, VK is a line perpendicular to ZZ’ and parallel to the x-axis. Then, the coordinates of focus S are (h, k) and the equation of the directrix ZZ’ is x = b. PM is perpendicular to directrix x = b and point M will be (b, y) Let us considered a point P(x, y) on the parabola. Now, join SP and PM. As we know that P lies on the parabola So, SP = PM (Parabola definition) SP2 = PM2 (x – h)2 + (y – k)2 = (x – b)2 + (y – y)2 x2 – 2hx + h2 + (y-k)2 = x2 – 2bx + b2 Add (2hx – b2) both side, we get x2 – 2hx + h2 + 2hx – b2 + (y-k)2 = x2 – 2bx + b2 + 2hx – b2 2(h – b)x = (y-k)2 + h2 – b2 Divide equation by 2(h – b), we get x =
Similarly when directrix y = b, we get
When V is origin, VS as x-axis of length a. Then, the coordinates of S will be (a, 0), and directrix ZZ’ is x = -a. h = a, k = 0 and b = -a Using the equation (1), we get x = x =
It is the standard equation of the parabola. Note: The parabola has two real foci situated on its axis one of which is the focus S and the other lies at infinity. The corresponding directrix is also at infinity. Tracing of the parabola y2 = 4ax, a>0 The given equation can be written as y = ± 2
Some other standard forms of the parabola with focus and directrixThe simplest form of the parabola equation is when the vertex is at the origin and the axis of symmetry is along with the x-axis or y-axis. Such types of parabola are: 1. y2 = 4ax Here,
2. x2 = 4ay Here,
3. y2 = – 4ay Here,
4. x2 = – 4ay Here,
Sample ProblemsQuestion 1. Find the equation of the parabola whose focus is (-4, 2) and the directrix is x + y = 3. Solution:
Question 2. Find the equation of the parabola whose focus is (-4, 0) and the directrix x + 6 = 0. Solution:
Question 3. Find the equation of the parabola with focus (4, 0) and directrix x = – 3. Solution:
Question 4. Find the equation of the parabola with vertex at (0, 0) and focus at (0, 4). Solution:
Focus & directrix of a parabola from the equationNow we will learn how to find the focus & directrix of a parabola from the equation. So, when the equation of a parabola is y – k = a(x – h)2 Here, the value of a = 1/4C So the focus is (h, k + C), the vertex is (h, k) and the directrix is y = k – C. Sample ExamplesQuestion 1. y2 = 8x Solution:
Question 2. y2 – 8y – x + 19 = 0 Solution:
Question 3. Find focus, directrix and vertex of the following equation: y = x2 – 2x + 3 Solution:
How do you find the equation of a parabola with the Directrix?The directrix of a parabola can be found, by knowing the axis of the parabola, and the vertex of the parabola. For an equation of the parabola in standard form y2 = 4ax, with focus at (a, 0), axis as the x-axis, the equation of the directrix of this parabola is x + a = 0 .
How do you find the equation of a parabola when given focus?Let (x0,y0) be any point on the parabola. Find the distance between (x0,y0) and the focus. Then find the distance between (x0,y0) and directrix. Equate these two distance equations and the simplified equation in x0 and y0 is equation of the parabola.
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Find the equation of the parabola with focus and directrix
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