## Focus question 4

In this page focus question 4 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. Since a is negative the parabola opens up in the left hand side.

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 4 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.