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).

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.

... The develoipment of analytic geometry and utilization of coordinates establishes the. ... The framework is called analytic geometry. It is based on coordinates. ...

... He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos, etc. as functions. ...

... Dynamic geometry offered the students the opportunity to use a computer to ... into new configurations as a challenge to their critical thinking and analytic skills ...

... Strand II. Geometry and Measurement. ... Standard IV.3 Number Relationships. Strand V. Numerical and Algebraic Operations and Analytical Thinking. ...

... of the mechanics for making fundamental domains and analytical solutions to the ... and 19th century were concerned with the calculus, geometry, arithmetic, number ...