If focus and directrix of a parabola are (3,-4) and x+y+7=0 then its length of latuusrectum is?
Need more clearly
The latus rectum's length is 2 times the distance from the focus to the directrix.
Now recall the point-to-line distance formula, and you get
|3*1-4*1+7|/√(1^2+1^2)
To find the length of the latus rectum of a parabola given its focus and directrix, follow these steps:
Step 1: Understand the concept
The latus rectum of a parabola is a line segment parallel to the directrix and passing through the focus. It is equal in length to the distance between the focus and any point on the parabola perpendicular to the directrix.
Step 2: Find the vertex
The vertex of a parabola is the midpoint between the focus and the directrix. Since the directrix equation is x + y + 7 = 0, we can rewrite it as y = -x - 7. The slope of this line is -1, and since the line is perpendicular to the latus rectum, the slope of the latus rectum will be 1.
From the equation of the directrix, we can find the x-coordinate of the vertex by setting y = -x - 7 equal to -4 (the y-coordinate of the focus):
-x - 7 = -4
-x = -4 + 7
-x = 3
x = -3
So the x-coordinate of the vertex is -3.
To find the y-coordinate, substitute the x-coordinate of the vertex into the equation of the directrix:
y = -(-3) - 7
y = 3 - 7
y = -4
Therefore, the vertex of the parabola is (-3, -4).
Step 3: Find the distance between the focus and the vertex
The distance between the focus and the vertex is the same as the distance between the vertex and the directrix. To find it, use the distance formula:
Distance = √((x₂ - x₁)² + (y₂ - y₁)²)
Using the coordinates of the focus (3, -4) and the vertex (-3, -4), the distance between them is:
Distance = √((-3 - 3)² + (-4 - (-4))²)
Distance = √((-6)² + 0²)
Distance = √(36 + 0)
Distance = √36
Distance = 6
Therefore, the distance between the focus and the vertex (and also the distance between the vertex and directrix) is 6.
Step 4: Find the length of the latus rectum
Since the latus rectum is a line segment that passes through the focus and is perpendicular to the directrix, it will be twice the distance between the focus and the directrix.
Length of latus rectum = 2 * Distance
Length of latus rectum = 2 * 6
Length of latus rectum = 12
Therefore, the length of the latus rectum of the given parabola is 12 units.