## Focus question 6

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.