Analytical geometry
1-Find the vertex focus, and directrex of the parabola=
1-x2= 2 (x+y)
2-Write an equation of the vertical parabola that contains (-1,2) with focus (3,1) and which opens upward.
3-Write an equation for the parabola that has vertex (1,2) and its axis is parallel to the x-axis, passes through the point (13,4)
4-Find all elements of the given ellipse and sketch the following equation: 25x2 + 9y2 - 100x +108 y + 199=0
5-Find the equation of the ellipse given that vertices (-8,0) and (8,0) with minor axis= 6
6-Write the equation of the ellipse given vertices are V (8,3) and V' (-4,3), one focus (6,3).
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Q1-find the vertex focus, and directrix of the parabolas =
Ans: 1-x2= 2(x+y) = 2x + 2y
Or x2 + 2x - 1= -2y
Or x2 + 2x + 1 - 2 = -2y
Or (x+1)2 = 2 - 2y = -4*0.5(y-1)
Or X2 = -4aY, where X = x+1, Y = y-1, a = 0.5
This is a parabola which opnes downwards with the vertex at (-1, 1).
The focus is at (0, -a) = (0, -0.5)
The directrix is y = a, or y = 0.5.
2-Write an equation of the vertical parabola that contains (-1,2) with focus (3,1) and which opens upward.
Ans: Since the parabola opens upwards, x should be squared and y should be linear.
If the equation of the parabola is of the type X2 = 4aY, then
Focus : X = 0, Y = a
Vertex: X = 0, Y = 0
Let X = x-h, Y = y-k, where the vertex is at (h, ...
Solution Summary
The expert finds the vertex focus and directrix of the parabola. Elements of the given ellipse of equations are found.
Analytical Geometry - Find the locus of a vertex.
The equation of the circle is given as:
x^2+y^2=25
A parallelogram is constructed as follows:
Vertex O is at the origin, vertex A is on the circle, vertex C is on the y-axis and the diagonal AC is parallel to the x-axis.
See attached figure
Question:
1. Find the locus of vertex B.
2. Describe the geometrical shape that is described by the equation you found in part 1, and find its intersection points with the axis.