How do you find the equation for this problem?
Conic Center(1,3) directrix x =7 snd corresponding focus (3,2)
To find the equation for this problem, you can use the standard form of the equation for a conic section known as a parabola:
(x-h)^2 = 4p(y-k)
where (h, k) is the vertex of the parabola and p is the distance from the vertex to the focus.
In this case, we are given the center of the conic as (1,3) and the corresponding focus as (3,2). Since the focus is not at the same x-coordinate as the vertex, we know that the parabola opens either to the left or right.
Next, we are given the directrix as x = 7. The directrix is always a vertical line that is equidistant from the vertex compared to the focus.
To find the value of p, we need to calculate the distance between the vertex and the focus. Distance between two points (x1, y1) and (x2, y2) is given by the distance formula:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, (x1, y1) = (1,3) and (x2, y2) = (3,2). Plugging in these values, we can calculate the distance, which is the value of p.
d = sqrt((3 - 1)^2 + (2 - 3)^2)
= sqrt(2^2 + (-1)^2)
= sqrt(4 + 1)
= sqrt(5)
Now that we have the value of p, we can substitute it into the standard form equation and use the given center to find the final equation.
Plugging in the values, we get:
(x - 1)^2 = 4(sqrt(5))(y - 3)
This is the equation for the parabola with the given characteristics.