Answer:
D
Step-by-step explanation:
Find the vertex
¼(x-5) = (y-2)²
x-5 = 4(y-2)²
x = 4(y-2)² + 5
This is a right-opening parabola with vertex (5,2).
focal length p = 1/(4·coefficient of y²) = 0.0625
directrix: x = 5-p = 4.9375
D
The equation of directrix of the parabola will be x = 4, i.e. Option A.
Directrix is a straight line distance to which from any point of a conic section is in fixed ratio to the distance from the same point to a focus. The directrix is perpendicular to the axis of symmetry and does not touch the parabola.
We have,
[tex]\frac{1}{4} (x - 5) = (y - 2)^2[/tex]
Now,
Rewrite as,
[tex](x - 5) = 4(y - 2)^2[/tex]
Now,
[tex]x= 4(y - 2)^2+5[/tex]
Now,
Comparing it with the vertex form,
i.e.
[tex]x = a ( y - h )^ 2 + k[/tex]
Vertex of parabola = (2, 5),
So,
Focus at (a, 0),
So,
Focus of the parabola = (4, 0),
i.e., a = 4
Now,
The Equation of directrix is given by x = a
So,
The equation of the directrix of this parabola,
x - a = 0,
i.e.
x - 4 = 0
⇒
x = 4
Hence, we can say that the equation of directrix of the parabola will be x = 4, i.e. Option A.
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