## Focus question 8

In this page focus question 8 we are going to find out the focus, vertex, equation of directrix and length of the latus rectum of the equation

x² +4y-6x+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. (y-k)²=4a(x-h)  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 +4y-6x+17=0. is in the quadratic equation form. Let us bring to the vertex form of equation.

x² +4y-6x+17=0.

x²-6x = -4y-17

x²-6x+9 = -4y-17+9(adding '1' on both sides)

(x-3) ² =  - 4y-8

(x-3) ² =   -4(y+2)

This is of the form (x-h)²=4a(y-k) whose vertex is (h,k)

Here h=3 and k=-2

and 4a = -4. So a = -4/4 =-1. Since a is negative, it opens down.

The focus is (h, k+a)  =  (3,-2-1) = (3,-3)

The vertex is (h,k)                    = (3,-2)

The equation of the directrix is y-k = -a

y-(-2)= -1

y=-3

The length of the latus rectum is 4a = 4

Parents and teachers help the students to solve the problem in the above method in focus question 8 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.