1: Length of lactus rectim of parabola with vertex(1,2) and directrix 3x-4y-2=0 is
A) 27/5 B) 18/7 C) 28/5
i kniw the formula to find length of lactus rectum =4a
The distance from (1,2) to the line 3x-4y-2=0 is
|3*1-4*2-2|/√(3^2+4^2) = 7/5
If the parabola is expressed in the form
y^2 = 4ax,
then the distance from the directrix to the vertex is a.
So, 4a = 28/5
Thank u soo mch fr helpin me out..
To find the length of the latus rectum of a parabola with a known vertex and directrix, you can follow these steps:
1. Recall that the standard equation for a parabola with its vertex at (h, k) and its directrix in the form ax + by + c = 0 is given by the equation: (x - h)^2 = 4a(y - k), where a represents the distance from the vertex to the focus and also the length of the latus rectum.
2. Start by determining the coordinates of the focus (h, k + a). In this case, the vertex is (1, 2), so to find the focus, add a to the y-coordinate: (1, 2 + a).
3. You know that the directrix is given by the equation 3x - 4y - 2 = 0. Convert the directrix equation into slope-intercept form (y = mx + b) to find the slope (m) and y-intercept (b).
4. Comparing the given equation 3x - 4y - 2 = 0 to the slope-intercept form, you can rearrange it as -4y = -3x + 2 and simplify it further as y = (3/4)x - 1/2. Thus, the slope (m) is 3/4 and the y-intercept (b) is -1/2.
5. Use the distance formula to find the length of the latus rectum. The distance formula for two points (x1,y1) and (x2,y2) is given by the equation: d = sqrt((x2 - x1)^2 + (y2 - y1)^2).
6. Substitute the coordinates of the focus and the intersection of the directrix and the axis of the parabola (for x1, y1 and x2, y2, respectively) into the distance formula.
7. The intersection point occurs when the directrix equation and the equation (x - h)^2 = 4a(y - k) are simultaneously true. Substituting the values of h, k, and a, solve for x.
8. Once you have the coordinates for the intersection point and the focus, substitute these values into the distance formula to find the length of the latus rectum (4a).
Following these steps, you can calculate the length of the latus rectum and identify the correct option from the given choices.