Analytic Geometry Answers

A(94,0,14) B(52,56,94) C(10,6,48) D(128,64,10) Content and instructions for completing task number 3.
The goal of the assignment is to apply methods of transforming orthogonal projections to solving metric problems.
The coordinates of four points ABCD are given (see next table). Determine the following values by transforming projection planes:
• 1. The size of the dihedral angle at the edge AB.
• 2. The shortest distance between ribs DA and BC.
• 3. Distance from vertex D to plane ABC.
• 4. Natural size of triangle ABC.
An example how to complete the task is shown after the table

find the equation of a parabola with a focus at (0,-1) and a directrix at y=4

The circle whose center is located in the first coordinate quarter touches the axis Ox at the point M, intersects two hyperbolas y=k1/x and y=k2/x (k1,k2 > 0) at the points A and B such that the line AB passes through the origin O. It is known that (4/k1) + (1/k2) = 20. Find the smallest possible length of the OM segment. In response, write down the square of the length of the segment OM.

Find the focus and equation of the latus rectum if x^2+ky=0, where k>0

The plane 3x+4y+2z=1 touches the conicoid 3x^2+2y^2+z^2=1
True or false with full explanation

There exists no line with 1/√3,1/√2, 1/√6 as direction cosines.
True or false with full explanation

Two stations, located at M(−1.5, 0) and N(1.5, 0) (units are in km), simultaneously send sound signals to a ship, with the signal traveling at the speed of 0.33 m/s. If the signal from N was received by the ship four seconds before the signal it received from M, find the equation of the curve containing the possible location of the ship.From the problem, how far in km are the two stations?
What is the equation of the curve containing the possible location of the ship?

Find the centre and radius of the equation of the circle. 2x^2+2y^2-8x+5y+10=0

12. What is the surface generated by a straight line that moves along a fixed curve, and which remains parallel to a fixed-line not on the curve?

Show that the cone whose vertex is at the origin and which passes through the curve vif intersection of the sphere x^2+y^2+z^2=3p^2 and any plane which is at the distance‘p' from the origin has mutually perpendicular generator s.