In this page focus question 2 we are going to find out the focus, vertex, equation of directrix and length of the latus rectum of the equation
y² -8x-2y+17=0.
Here the equation is in the standard form (y-k)²=4a(x-h) .The following table gives the necessary details of the standard and vertex form of parabola.
Standard form |
Vertex form |
y² =4ax If a is positive, then it opens in the right hand side. If a is negative, then it opens in the left hand side. The focus is (a,0). The vertex is the origin (0,0) The equation of the directrix is x =-a The length of the latus rectum is 4a. |
(y-k)²=4a(x-h) If a is positive, then it opens in the right hand side. If a is negative, then it opens in the left hand side. The focus is (h+a, k) The vertex is (h,k) The equation of the directrix is x-h = -a The length of the latus rectum is 4a. |
Solution:
Here the equation y² -8x-2y+17=0 is in the quadratic equation form. Let us bring to the vertex form of equation.
y² -8x-2y+17=0.
y²-2y = 8x-17
y²-2y+1 = 8x-17+1(adding '1' on both sides)
(y-1) ² = 8x-16
(y-1) ² = 8(x-2)
This is of the form (y-k)²=4a(x-h) whose vertex is (h,k)
Here h=2 and k=1
and 4a = 8. So a = 8/4 =2.
The focus is (h+a, k) = (2+2,1) = (4,1)
The vertex is (h,k) = (2,1)
The equation of the directrix is x-2 = -2
x=0
The length of the latus rectum is 4a =8
Parents and teachers help the students to solve the problem in the above method in focus question 2 and they can guide them to solve the following problem using the above method.
The other three standard forms and vertex forms of parabola are discussed in the focus worksheet.
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Problem for practice: