Respuesta :
Answer:
(x - 8)² = 20(y + 6)
Step-by-step explanation:
The focus of the parabola is (8,-1) and the equation of the directrix is y = - 11.
So, the parabola opens up and the axis of symmetry is x = 8 line.
Now, the axis of symmetry and the directrix meet at the point (8,-11).
So, the vertex of the parabola will be the midpoint of (8,-1) and (8,-11) i.e. (8,-6).
Now, half of the distance between the focus and the directrix is = a = [tex]\frac{11 - 1}{2} = 5[/tex] units.
So, the equation of the parabola in vertex form will be
(x - 8)² = 4 × 5 × (y + 6)
⇒ (x - 8)² = 20(y + 6) (Answer)
The equation of parabola of given properties is:
[tex]x^2 - 16x = 20y + 56[/tex]
Given data about parabola:
Coordinates of point of focus: (8, -1)
Equation of directrix: y = -11
The key fact in solving such problem is that each point of parabola is equally distant from the point of focus and perpendicular line on the directrix.
Let a point of parabola be [tex](x_0, y_0)[/tex]
Then, we have:
[tex]\text{Distance of} \: (x_0, y_0) \text{ \:\rm from \:} (8,-1)= \text{Distance from point on perpendicular on directrix}\\or\\\sqrt{(x_0-8)^2 + (y_0 -(-1))^2} = \sqrt{(x_0-x_0)^2 + (y_0 -(-11))^2}\\\\\\x_0^2 + 64 -16x_0 + y_0^2 + 1 + 2y_0 = y_0^2 + 121 + 22y_0\\x_0^2 -16x_0 = 20y_0 + 56\\\\[/tex]
Since this is true for all (x,y), thus we get equation of parabola as:
[tex]x^2 - 16x = 20y + 56[/tex]
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