1. Given the parabola defined by y^2 =-12x, find the coordinates of the focus,the length of the latus rectum and the coordinates of its endpoints.Find also the equation of the directrix. Sketch the curve.
2. Find the eccentricity of an ellipse when the length of the latus rectum is 2/3 of the major axis.
3. Plot the curve
x = 3-cos they a, y= 2+sin theta
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1. To find the coordinates of the focus, length of the latus rectum, endpoints, and equation of the directrix, we can use the standard form equation of a parabola: y^2 = 4ax, where (h, k) is the vertex and a is the distance between the vertex and focus/directrix.
Comparing this to the given equation y^2 = -12x, we can see that 4a = -12, so a = -3.
- The vertex can be found by setting y = 0 in the given equation:
0^2 = -12x
h = 0
Therefore, the vertex is (0, 0).
- The distance between the vertex and focus is equal to a. So, the coordinates of the focus are (a, 0) = (-3, 0).
- The length of the latus rectum is given by 4a, which is 4(-3) = -12.
- To find the coordinates of the endpoints, we can use the fact that the endpoints are equidistant from the vertex.
The distance from the vertex to an endpoint is also equal to a. So, we can calculate the x-coordinate of the endpoints as h + a = 0 + (-3) = -3.
Plugging this x-coordinate back into the equation, we get y = ±√(-12x) = ±√(36) = ±6.
Therefore, the two endpoints are (-3, 6) and (-3, -6).
- The equation of the directrix can be found by using the formula x = -a:
x = -(-3), which simplifies to x = 3.
To sketch the curve, plot the vertex (0, 0), focus (-3, 0), and endpoints (-3, 6) and (-3, -6) on a graph. The directrix is a vertical line passing through x = 3. Draw a smooth curve passing through the vertex, focus, and endpoints, curving towards the directrix.
2. The latus rectum of an ellipse is defined as 2b^2/a, where a is the length of the major axis and b is the length of the minor axis.
Given that the latus rectum is 2/3 of the major axis, we can set up the equation:
2/3 = 2b^2/a
Simplifying, we get:
2a = 3b^2
To find the eccentricity (e) of the ellipse, we can use the formula:
e = √(1 - (b^2/a^2)
Since we know the relationship between a and b, we can substitute it into the formula:
e = √(1 - (1.5/a)
3. To plot the curve x = 3 - cos(θ) and y = 2 + sin(θ), we can assign different values to the variable θ and calculate the corresponding x and y values.
- Choose a set of values for θ, such as θ = 0, π/4, π/2, 3π/4, π, etc.
- Plug each θ value into the equations x = 3 - cos(θ) and y = 2 + sin(θ).
- Calculate the corresponding x and y values for each θ.
- Plot the points (x, y) on a graph.
- Connect the plotted points to form a smooth curve.
By following these steps, you can obtain the plot of the given curve.