what will be equation of latus rectum of parabola x^2-2x+4y=0.
First, the definition of latus rectum of a parabola is the line segment perpendicular to the axis, passing through the focus of the parabola and which has its two end-points on the curve.
Refer to the illustration below:
http://imageshack.us/photo/my-images/195/1312812014.png/
The focus is at (1,-3/4) through which the red line passes. The Latus rectum is the red line segment on the interior of the curve (from -1 to 3).
To find the latus rectum, first reduce the given parabola to the form:
y=(1/4p)(x-h)^2+k
using completing the squares, and where (h,k) is the vertex. The focus is at
(h,k+p) the length of the latus rectum is 4p.
The focus is at a distance p from the vertex, interior to the curve.
For the given parabola,
x^2-2x+4y=0
reduces to
f(x)=y=-(1/4)(x^2-2x)
=-(1/4)(x-1)^2+1/4
=(1/(4(-1)))(x-1)^2+1/4
Thus the vertex is at (1,1/4) and p=-1
The focus is at (1,1/4+p)=(1,-3/4)
Since
f(-1)=-3/4, and f(3)=-3/4
we conclude that the latus rectum is between the points
(-1,-3/4) and (3,-3/4), and of length 4 (equal to 4p).
Given: x^2 - 2x + 2y = 0.
4y = -x^2 + 2x,
y = -(1/4)x^2 + (1/2)x,
a = -1/4,
LR = [1/a] = 4 = Latus Rectum.
4a = 4(-1/4) = -1.
1/4a = -1.
[1/4a] = 1.
h = Xv = -b/2a = (-1/2) / (-1/2) = 1.
k = Yv = -1^2/4 + (1/2)1 = 1/4.
V(1,1/4)
F(1,y)
Y = k - [1/4a] = 1/4 --1 = -3/4.
F(1,-3/4)
Eq of LR: Y = -3/4.
To find the equation of the latus rectum of the parabola, we need to determine the coordinates of the focus and the length of the latus rectum.
The given equation of the parabola is x^2 - 2x + 4y = 0. We can rewrite it in standard form as y = (1/4)(x^2 - 2x).
Comparing this with the standard form of a parabola, which is y = 4a(x - h)^2 + k, we can determine the values of h and k:
h = 2
k = 0
The vertex form of the parabola is given by (h, k), so the vertex of this parabola is V(2, 0).
Since the coefficient of x in the equation is 1, the focal length of the parabola is given by 1/4a, where a is the coefficient of x^2. In this case, a = 1, so the focal length is 1/4.
Using the vertex and the focal length, we can find the coordinates of the focus F, which can be found by adding the focal length to the x-coordinate of the vertex. Therefore, the focus F is located at F(2 + 1/4, 0) = F(9/4, 0).
Now, we can find the equation of the latus rectum using the focus F. The latus rectum is a line segment passing through the focus and perpendicular to the axis of the parabola.
Since the parabola opens upwards or downwards, the latus rectum is parallel to the x-axis. The length of the latus rectum is twice the focal length, which is 2(1/4) = 1/2.
Therefore, the equation of the latus rectum is x = 9/4.
In summary, the equation of the latus rectum of the parabola x^2 - 2x + 4y = 0 is x = 9/4.
To find the equation of the latus rectum of a parabola, we first need to express the given parabola in the standard form of a parabola, which is of the form y^2 = 4ax.
Given equation: x^2 - 2x + 4y = 0
Step 1: Complete the square for the x terms.
x^2 - 2x = -4y
x^2 - 2x + 1 = -4y + 1, adding 1 to both sides to complete the square
(x - 1)^2 = -4y + 1
Step 2: Rearrange the equation to match the standard form.
-4y + 1 = (x - 1)^2
-4y = (x - 1)^2 - 1, subtracting 1 from both sides
y = (-1/4)(x - 1)^2 + 1/4, dividing by -4 and rearranging terms
Now, we can see that the standard form of the parabola is y^2 = 4ax with a = -1/4.
The latus rectum of a parabola with the parameter a is given by the equation x = a.
In this case, a = -1/4, so the equation of the latus rectum is x = -1/4.
Therefore, the equation of the latus rectum of the given parabola x^2 - 2x + 4y = 0 is x = -1/4.