Basic Properties of Second-Order Surfaces

Find the center and radius of the circle

Write the equations of the planes passing through the line

and tangent to the sphere $x^{2}+y^{2}+z^{2}-2 x-4 y-6 z-67 = 0$.

Find the equation of the sphere passing through the circle

and through the origin.

Prove that the line of intersection of the two parabolic cylinders $y^{2} = x, z^{2} = 1-x$ lies on a circular cylinder. What is the equation of this cylinder?

Prove that a paraboloid of revolution and a circular cylinder whose axes are parallel intersect in an ellipse whose major axis lies in the plane passing through the axes of the given surfaces.

Along what curve do the one-sheeted hyperboloid $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}-\frac{z^{2}}{c^{2}} = 1, a > b$, and the sphere $x^{2}+y^{2}+z^{2} = a^{2}$ intersect?

Prove that the line of intersection of the parabolic cylinder $z^{2} = x+y$ with the paraboloid of revolution $x^{2}+y^{2} = z$ lies on a sphere. Find the center and radius of this sphere.

Along what curve do the elliptic paraboloid $\frac{x^{2}}{p}+\frac{y^{2}}{q} = 2 z, p > q > 0$, and the sphere $x^{2}+y^{2}+z^{2} = 2 p z$ intersect?

Prove that the orthogonal projection of any plane section of a paraboloid of revolution onto a plane perpendicular to the axis of the paraboloid is a circle or a line.

Along what curve do the two ellipsoids

intersect?

Find the line of intersection of the surfaces $x^{2}+y^{2}-z^{2} = a^{2}$, $x^{2}-y^{2} = 2 a z$.

Find the acute angle between the generators of the cone $x^{2}+y^{2}- -z^{2} = 0$ along which it is intersected by the plane $5 x+10 y-11 z = 0$.

Determine the type and position of the line of intersection of the plane $x = 9$ and the hyperboloid

Write the equation of the plane parallel to the plane $O y z$ and intersecting the one-sheeted hyperboloid

in a hyperbola whose real semi-axis equals 1.

Find the necessary and sufficient condition for the plane $z = a x+b y+c$ to intersect the paraboloid of revolution $x^{2}+y^{2} = 2 p z(p > 0)$ in a real ellipse.

Along what curve do the hyperbolic paraboloid

and the plane $2 x+3 y-6 = 0$ intersect?

Find the equation of a second-order surface, given that it intersects the plane $O x y$ in the circle $x^{2}+y^{2}-12 x- -18 y+32 = 0, z = 0$, and the planes $O x z$ and $O y z$ in parabolas whose axes are parallel to the positive direction of the axis $O z$, with the parameter of the parabola lying in the plane $O x z$ equal to 1.

Write the equation of a second-order surface intersecting the plane $O x y$ in a parabola, and the planes $O x z$ and $O y z$ in circles of radius $r$ tangent to the positive coordinate semi-axes. Using a transformation of rectangular coordinates, reduce the equation to canonical form.

Find the locus of foci of the hyperbolas obtained by intersecting the hyperbolic paraboloid

with planes parallel to the plane Oxy.

Find the equation of a second-order surface passing through the points $(0,0,0),(1,1,-1),(0,0,1)$, for which the planes

are planes of symmetry. Write the canonical equation of this surface and indicate its type.

Find the equation of a second-order surface for which the planes

are planes of symmetry and which passes through the points $(1,0,0),(0,-1,0),(1,1,-1)$.

Find the type of surface consisting of the lines along which mutually perpendicular planes passing through two given lines intersect, depending on the relative position of the latter.

Find the locus of points such that the ratio of the distances from them to two given skew lines is the same number $k$.