parabola y^2=px pass through (4,-2).find the latus rectum and focus.
To find the latus rectum and focus of the parabola y^2 = px that passes through the point (4, -2), we can use the properties of the parabola and the given information.
First, let's understand the equation of the parabola y^2 = px. This is a standard equation of a parabola with its vertex at the origin (0, 0), axis along the y-axis, and focus at the point (p/4, 0). The value of p determines the position of the vertex and the focus.
Now, we have the point (4, -2) that lies on the parabola. Let's substitute these coordinates into the equation y^2 = px:
(-2)^2 = p(4)
4 = 4p
p = 1
So, from this, we can conclude that the value of p is 1.
Now, let's find the focus of the parabola. We know that the focus is located at (p/4, 0). Substituting the value of p = 1, we get:
Focus = (1/4, 0)
Therefore, the focus of the parabola is (1/4, 0).
Next, let's find the latus rectum of the parabola. The latus rectum is a line segment passing through the focus, perpendicular to the axis of the parabola, and with endpoints on the parabola.
The length of the latus rectum can be calculated using the formula:
Latus rectum = 4p
Substituting the value of p = 1, we get:
Latus rectum = 4(1)
Latus rectum = 4
Therefore, the length of the latus rectum is 4 in this case.
In conclusion, for the parabola y^2 = px that passes through the point (4, -2), the focus is at (1/4, 0), and the length of the latus rectum is 4.