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. Show
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:
What is parabola using focus and Directrix?What are the focus and directrix of a parabola? Parabolas are commonly known as the graphs of quadratic functions. They can also be viewed as the set of all points whose distance from a certain point (the focus) is equal to their distance from a certain line (the directrix).
How do I find the equation of a parabola?How to find a parabola's equation using its Vertex Form. Step 1: use the (known) coordinates of the vertex, (h,k), to write the parabola's equation in the form: y=a(x−h)2+k. ... . Step 2: find the value of the coefficient a by substituting the coordinates of point P into the equation written in step 1 and solving for a.. What are the equation of Directrices of parabola?
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