Analytic Geometry Answers

Find the equations of the spheres which pass through the circle x²+y²+z²=9, 2x+2y-7=0 and touch the plane x-y+z+3=0

Obtain the equation of the conic, a focus of which lies at (2,1), the directrix of which is x+y=0 and which passes through (1,4). Also identify the conic.

Q.Choose the correct answer.
Q. The equation 2x2+7y2-4xy+3y-1=0 represents
a. A pair of straight lines
b. Ellipse
c. Parabola
d. Hyperbola

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.