In this page focus question 6 we are going to find out the focus, vertex, equation of directrix and length of the latus rectum of the equation
x² -8y-2x+17=0.
Here the equation is in the standard form (x-h)²=4a(y-k).The following table gives the necessary details of the standard and vertex form of parabola.
Standard form |
Vertex form |
x² =4ay If a is positive, then it opens up . If a is negative, then it opens down. The focus is (0,a). The vertex is the origin (0,0) The equation of the directrix is y =-a The length of the latus rectum is 4a. |
(x-h)²=4a(y-k) If a is positive, then it opens up . If a is negative, then it opens down. The focus is (h, k+a) The vertex is (h,k) The equation of the directrix is y-k = -a The length of the latus rectum is 4a. |
Solution:
Here the equation x² -8y-2x+17=0 is in the quadratic equation form. Let us bring to the vertex form of equation.
x² -8y-2x+17=0.
x²-2x = 8y-17
x²-2x+1 = 8y-17+1(adding '1' on both sides)
(x-1) ² = 8y-16
(x-1) ² = 8(y-2)
This is of the form (x-h)²=4a(y-k) whose vertex is (h,k)
Here h=1 and k=2
and 4a = 8. So a = 8/4 =2.
The focus is (h, k+a) = (1,2+2) = (1,4)
The vertex is (h,k) = (1,2)
The equation of the directrix is y-k = -2
y-2= -2
y=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 6 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: