Analytic Geometry Answers

The plane lx+my+nz=0 moves in such a way that it's intersection with the planes ax+by+cz+d=0 , a'x+b'y+c'z+d'=0 are perpendicular. Show that the normal to the plane through the origin describes, in general, a cone of second degree , and find its equation . Examine the case when aa'+bb'+cc'=0 , that is , when the two given planes are perpendicular.

Tangent plane at any point of sphere x^2+y^2+z^2=r^2 meets the coordinate axis at A,B, C. Show that the locus of the point of intersection of planes drawn parallel to the coordinate planes through A,B,C is the surface x^-2+y^-2+z^-2=r^-2

Obtain the equation of spheres that pass through the points (4,1,0), (2,-3,4), (1,0,0) and touch the plane 2x+2y-z=11

Find the centres of the two spheres which touch the plane 4x+3y=47 at the point (8,5,4) and the sphere x^2+y^2+z^2=1

Find the equation of the spheres of radius r which touch the three coordinate axis . How many such sphereshort are there?

Find the equation of tangent line to the circle x^2+y^2+z^2+5x-7y+2z-8=0,3x-2y+4z+3=0 and the point (-3,5,4)

Chanelle is creating a design for vinyl flooring.

She uses circles and squares to create the design shown.

If the equation of the small circle is x2 + y2 = 16, what are the dimensions of the large square?

Ok, so, in the cartesian plane, the order of the shapes is as follows:

small circle

small square

larger circle

larger square

Find an equation of the ellipse that has a center(0,5), a minor axis of length 8, and a vertex at  (0,-1).

Identify the conic x^2- 2xy + y^2 + √2x = 2

 Draw the hyperbola with the equation: 3𝑥 2 − 2𝑦 2 + 6𝑥 − 8𝑦 = 6