Question -
Answer -
(i)┬аy2┬а= 8x
Given:
Parabola = y2┬а= 8x
Now by comparing with the actual parabola y2┬а=4ax
Then,
4a = 8
a = 8/4 = 2
So, the vertex is (0, 0)
The focus is (a, 0) = (2, 0)
The equation of the axis is y = 0.
The equation of the directrix is x = тАУ a i.e., x = тАУ 2
The length of the latus rectum is 4a = 8.
(ii)┬а4x2┬а+ y = 0
Given:
Parabola => 4x2┬а+ y = 0
Now by comparing with the actual parabola y2┬а=4ax
Then,
4a = ┬╝
a = 1/(4 ├Ч 4)
= 1/16
So, the vertex is (0, 0)
The focus is = (0, -1/16)
The equation of the axis is x = 0.
The equation of the directrix is y = 1/16
The length of the latus rectum is 4a = ┬╝
(iii)┬аy2┬атАУ 4y тАУ 3x + 1 = 0
Given:
Parabola y2┬атАУ 4y тАУ 3x + 1 = 0
y2┬атАУ 4y = 3x тАУ 1
y2┬атАУ 4y + 4 = 3x + 3
(y тАУ 2)2┬а= 3(x + 1)
Now by comparing with the actual parabola y2┬а=4ax
Then,
4b = 3
b = ┬╛
So, the vertex is (-1, 2)
The focus is = (3/4 тАУ 1, 2) = (-1/4, 2)
The equation of the axis is y тАУ 2 = 0.
The equation of the directrix is (x тАУ c) = -b
(x тАУ (-1)) = -3/4
x = -1 тАУ ┬╛
= -7/4
The length of the latus rectum is 4b = 3
(iv)┬аy2┬атАУ 4y + 4x = 0
Given:
Parabola y2┬атАУ 4y + 4x = 0
y2┬атАУ 4y = тАУ 4x
y2┬атАУ 4y + 4 = тАУ 4x + 4
(y тАУ 2)2┬а= тАУ 4(x тАУ 1)
Now by comparing with the actual parabola y2┬а=4ax => (y тАУ a)2┬а= тАУ 4b(x тАУ c)
Then,
4b = 4
b = 1
So, the vertex is (c, a) = (1, 2)
The focus is (b + c, a) = (1-1, 2) = (0, 2)
The equation of the axis is y тАУ a = 0 i.e., y тАУ 2 = 0
The equation of the directrix is x тАУ c = b
x тАУ 1 = 1
x = 1 + 1
= 2
Length of latus rectum is 4b = 4
(v)┬аy2┬а+ 4x + 4y тАУ 3 = 0
Given:
The parabola y2┬а+ 4x + 4y тАУ 3 = 0
y2┬а+ 4y = тАУ 4x + 3
y2┬а+ 4y + 4 = тАУ 4x + 7
(y + 2)2┬а= тАУ 4(x тАУ 7/4)
Now by comparing with the actual parabola y2┬а=4ax => (y тАУ a)2┬а= тАУ 4b(x тАУ c)
Then,
4b = 4
b = 4/4 = 1
So, The vertex is (c, a) =┬а(7/4, -2)
The focus is (- b + c, a) = (-1 + 7/4, -2) = (3/4, -2)
The equation of the axis is y тАУ a = 0 i.e., y + 2 = 0
The equation of the directrix is x тАУ c = b
x тАУ 7/4 = 1
x = 1 + 7/4
= 11/4
Length of latus rectum is 4b = 4.