For a parabola; the equation of the directrix is: x + y = 4 and the coordinates of focus F: (6,6). Find

(a) Vertex

(b) Axis of parabola

Hint: Here, from the above figure we came to know that Axis of the parabola is perpendicular to the directrix and passes through the focus of the parabola. Now, analyse the given and proceed to solve the rest of the solution.

Complete step by step answer:

As, equation of the directrix is x + y = 4.

$\Rightarrow$ x + y = 4 - (1)

Slope of the directrix will be ${m_1} = - 1$.

As, directrix and axis are perpendicular so form ${m_1}{m_2} = - 1$

Where ${m_2}$ is the slope of the axis of the parabola.

$\Rightarrow {m_2} = 1$

So, the equation of axis of parabola with slope 1 and passing thous focus (6,6) will be,

$\Rightarrow (y - 6) = 1(x - 6)$

Equation of axis of parabola will be x - y = 0

$\Rightarrow$ x - y = 0 - (2)

As, from the above figure we see that vertex will always be the midpoint of focus of parabola and intersection of axis and directrix of parabola.

So, the intersection of the axis and the directrix of the parabola will be C (x,y).

Now, solving equation 1 and 2 for point C (x,y).

Adding equation 1 and 2 we get,

$\Rightarrow$ 2x = 4

$\Rightarrow$ x = 2

Putting value of x in Equation 2 we get,

$\Rightarrow$ y = 2

So, C $\equiv \left( {2,2} \right)$

Now, as we can see from the above figure vertex will be the mid point of point C and focus of parabola.

So, Vertex $\equiv \left( {\frac{{6 + 2}}{2},\frac{{6 + 2}}{2}} \right) \equiv \left( {4,4} \right)$

NOTE: - All parabola have the same properties only their vertex and axis shift. So, we can always find the vertex and axis of a parabola by considering the properties of the standard parabola. If not asked then there is no need to find an equation of parabola.