Analytic Geometry Answers

Examine which of the following conicoids are central and which are non-central.
Also detrrmine which of the central conicoids have centre at the origin.
1. x²+y²+z²+x+y+z=1
2.2x²+4xy+xz-x-3y+5z+3=0
3.x²-y²-z²+xy+4yz+x=0

Prove that the plane 2x-3y+6z=6 touches the conicoid 4x²-9y²+36z²=36
Find the point of contact.

Identify and trace the conic x²-2xy+y²-3x+2y+3=0

Find the equation of the normal to the parabola y²+4x=0 at the point where the line y=x+c touches it.

Q.Choose the correct answer.
Q. The points (1/√3 , 1), (2/√3, 2),(1/√3,3) are the vertices of
a) Isosceles triangle
b) Equilateral
c) Right Triangle
d) None of above

determine the coordinates of P and Q if the sketch shows the hyperbola defined by y=4/9x;the straight line defined by y=x;a circle with a centre at P,touching the x-axis and y-axis at R and S ,respectively ;and the straight line through the point T,S and R. The line joining T and Q is parallel to the y-axis

Assume there are two planes Π1 and Π2 for which there exists a line of intersection L.
a) Suppose you find two different parametric forms for L using two different methods. How can you prove they are the same line?
b) Let there be vectors n1 and n2 that are normals to Π1 and Π2 respectively. If m = n1×n2 (cross product) and m is parallel to L, give a geometric description of this result.

Find the vertex, focus, and Directrix of each parabola.
1. (x+2)^2=20(y+5)
2. (y-1)^2=-16(x+4)
3. (x-5)^2=4(y+1)
4.(y-3)^2=12(x+1)
5. (x+4)^2=6(y-5)

An archeologist found the remains of an ancient wheel, which she then placed on a grid. If an arc of the wheel passes through A(–7, 0), B(–3, 4) and C(7, 0), locate the center of the wheel, and the standard equation of the circle defining its boundary.

Find the vector equation of the line passing through 2i - 3j and i + j + k. Which point on this line has direction cosines, 1/√3, 1/√3, 1/√3 and why?