# 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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#### Solution Preview

Please see attached file.

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.