Question

Write down the equation of given parabola x? +8x+4y+12 =0 in standard form. State the vertex, focus and the equation of the directrix. Hence, sketch its graph.

Answer 1

6. Given equation is . Simplifying the equation, we get:

x2 + 8x + 4y + 12 = 0

12 + 8x + 4y + 12 = 0 ►r? +2..4+ 42 - 42 + 4y + 12 = 0 = (x + 4) = -4y + 4 = (x + 4) = -4(y - 1)

From the above standard form, we get to know that it is a downward opening parabola. letting X=x+4 and Y=y-a, we get the general form as X²= 4aY, with a= - 1.

For the vertex, we have X=0 and Y=0. So x+4=0 and y-1=0. Solving, we get, x=-4 and y=1. Hence, the vertex lies at (-4,1).

The focus lies at X=0 and Y=a. Thus, x+4=0 and y-1= -1. Solving, x= -4 and y= 0. Hence, the focus lies at (-4,0).

The equation of the directrix is given by Y= -a or y-1= -(-1) or y-1=1. Thus the equation of directrix is y=2.

For better sketching we find where the curve cuts x-axis by putting y=0 and solving:

(x + 4) = -4(y - 1) (x + 4) = -4(0 - 1) (x + 4) = 4 r+4= 2 or r +4= -2 =r= -2 or r = -6

Hence, it cuts x-axis at (-2,0) and (-6,0).

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