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Question: If (0,4) and (0,2) are respectively the vertex and focus of a parabola, then its equation is A) \[...

If (0,4) and (0,2) are respectively the vertex and focus of a parabola, then its equation is
A) x2+8y=32x^{2}+8y=32
B) y2+8x=32y^{2}+8x=32
C) x2−8y=32x^{2}-8y=32
D) y2−8x=32y^{2}-8x=32

Explanation

Solution

Hint: In this question, first draw the diagram of parabola it will give us a clear picture of what we have to find out, while drawing we have to keep in mind that Vertex(0,4) and focus S(0,2) lies on X-axis, so the axis of this parabola is must be Y-axis and this parabola opens downward(i.e negative Y-axis).

After that we will apply the formula-

Equation of a parabola: (x−α)2=−4a(y−β)\left( x-\alpha \right)^{2} =-4a\left( y-\beta \right) ............equation(1)

Where (α,β)\left( \alpha ,\beta \right) Vertex of this parabola and a is the distance between focus and vertex.

Complete step by step answer:

In this question it is given that the vertex of this parabola is (0,4) and also by equation(1) we can say that the value of (α,β)\left( \alpha ,\beta \right) is (0,4).

So by putting the value in equation(1) we get,

(x−0)2=−4a(y−4)\left( x-0\right)^{2} =-4a\left( y-4\right) ……...equation (2)

Now we have to find the value of a. and since a is the distance between focus (0,2) and vertex (0,4).

For this we have to know the distance formula i.e, the distance between the points (a,b) to (c,d) is d=(a−c)2+(b−d)2\mathbf{d} =\sqrt{\left( a-c\right)^{2} +\left( b-d\right)^{2} } ............equation( 3)

Now by using equation(3) we can find,

a=(0−0)2+(4−2)2a=\sqrt{\left( 0-0\right)^{2} +\left( 4-2\right)^{2} }=22\sqrt{2^{2}} =2

Now we have got the value of a, so by putting the value of a=2 in equation(2) we get,

(x−0)2=−4×2(y−4)\left( x-0\right)^{2} =-4\times 2\left( y-4\right)

⇒x2=−8(y−4)\Rightarrow x^{2}=-8\left( y-4\right)

⇒x2=−8y+32\Rightarrow x^{2}=-8y+32

⇒x2+8y=32\Rightarrow x^{2}+8y=32

So this is the required equation of parabola.

Hence the correct option is option A.

Note: In this type of question we generally forgot to identify the axis of parabola and the direction of its open face and because of that we end up taking a wrong equation of parabola, so for this you first recognised that the axis of the parabola is parallel to which axis(X or Y - axis) and after that the direction of its open face.