If (0,4) and (0,2) are respectively the vertex and focus of a parabola, then its equation is
A) $$x^{2}+8y=32$$
B) $$y^{2}+8x=32$$
C) $$x^{2}-8y=32$$
D) $$y^{2}-8x=32$$
VerifiedHint: 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: $$\left( x-\alpha \right)^{2} =-4a\left( y-\beta \right)$$ ............equation(1)
Where $\left( \alpha ,\beta \right)$ 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 $\left( \alpha ,\beta \right)$ is (0,4).
So by putting the value in equation(1) we get,
$$\left( x-0\right)^{2} =-4a\left( y-4\right) $$ ……...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 $\mathbf{d} =\sqrt{\left( a-c\right)^{2} +\left( b-d\right)^{2} }$ ............equation( 3)
Now by using equation(3) we can find,
$$a=\sqrt{\left( 0-0\right)^{2} +\left( 4-2\right)^{2} }$$=$$\sqrt{2^{2}}$$ =2
Now we have got the value of a, so by putting the value of a=2 in equation(2) we get,
$$\left( x-0\right)^{2} =-4\times 2\left( y-4\right) $$
$$\Rightarrow x^{2}=-8\left( y-4\right) $$
$$\Rightarrow x^{2}=-8y+32$$
$$\Rightarrow x^{2}+8y=32$$
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.
