Answer:
(y - 2)² = -12(x - 5)
Explanation:
A parabola is a locus of points, which are equidistant from the focus and directrix;
Generic cartesian equation of a parabola:
y² = 4ax, where the:
Focus, S, is: (a, 0)
Directrix, d, is: x = -a
a > 0
Put simply, a is the horinzontal difference between the directrix and the vertex or between the vertex and focus;
Always a good idea to do a quick drawing of the graph;
We are the told the focus, F, is: (2, 2) and directrix, d, is: x = 8;
First thing to note, the vertex, or turning point will be in line with the focus vertically, i.e. they will share the same y-coordinate;
Horizonatally, it will be halfway between the focus and the directrix, i.e. halfway between 8 and 2;
Therefore, the vertex will be will be (5, 2);
We can also work out a:
a = 8 - 5 = 5 - 2
a = 3
Substituting this value of a into the generic cartesian equation:
y² = 4(3)x
y² = 12x
The focus and directrix will be:
S: (3, 0)
d: x = -3
Next thing to note, a parabola curves away from the directrix;
In this case, the directrix is x = 8, so the vertex will be the right-most point on the parabola, it will curve off to the left and the focus will also be to the left;
What we want to do is compare with y² = 12x;
This parabola, has a vertex (0, 0), which is the left-most point that curves off to the right and a focus also to the right;
Since we know the formula of this parabola, if we figure out how to transform it into the one in the question, we can find out it's equation;
What we should recognise first is that the parabola in the question is reflected in the y-axis, compared to y² = 12x;
So we apply the transformation that corresponds to this, i.e. use the f(-x) rule:
y² = 12(-x)
y² = -12x
Now the two graphs will have the same shape and orientation;
The focus and directrix will also be affected:
S: (-3, 0)
d: x = 3
Now, the only remaining difference would be the coordinates of the focus and directrix of the two graphs;
The focus of the graph in the question is 5 units to the right and 2 units upwards compared to the focus of y² = -12x;
The directrix is 5 units to the right of that of y² = -12x;
So we apply a translation transformation of 5 units right and 2 units up, like so:
(y - 2)² = -12(x - 5)
Replace y with (y - 2) to translate up 2 units;
Replace x with (x - 5) to translate 5 units right.
We know have a parabola with focus, (2, 2), directrix, x = 8 and vertex, (5, 2), i.e. the parabola in the question;
Hence, the equation of the parabola in the question is:
(y - 2)² = -12(x - 5)
It might seem a bit long and complicated to begin with, but can be done very quickly if you can get used to it.