Tangent lines and normal planes of space curves

Write the tangent line and normal plane equations at the given point for the following curve: $x=a\cos \alpha \cos t$, $y=a\sin \alpha \cos t$, $z=a\sin t$; at the point $t=t_{0}$.

Write the tangent line and normal plane equations at the given point for the following curve: $x=a\sin^{2}t$, $y=b\sin t\cos t$, $z=c\cos^{2}t$; at the point $t=\dfrac {\pi }{4}$.

Write the tangent line and normal plane equations at the given point for the following curve: $y=x$, $z=x^{2}$; at the point $M(1,1,1)$.

Write the tangent line and normal plane equations at the given point for the following curve: $x^{2}+z^{2}=10$, $y^{2}+z^{2}=10$; at the point $M(1,1,3)$.

Write the tangent line and normal plane equations at the given point for the following curve: $x^{2}+y^{2}+z^{2}=6$, $x+y+z=0$; at the point $M(1,-2,1)$.

Find a point on the curve $x=t$, $y=t^{2}$, $z=t^{3}$ at which the tangent line is parallel to the plane $x+2y+z=4$.

Prove that the tangent line to the helix $x=a\cos t$, $y=a\sin t$, $z=bt$ makes a constant angle with the $Oz$ axis.

Prove that the curve

intersects all the generating lines of the cone $x^{2}+y^{2}=z^{2}$ at the same angle.

Prove that the loxodrome

(where $\varphi$ is the longitude of a point on the Earth, and $\psi$ is the latitude of a point on the Earth) intersects all meridians of the Earth at a constant angle.

Given the curve

where $f$ is a differentiable function. Find the tangent of the angle formed by the tangent line to the curve at the point $M_{0}(x_{0}, y_{0})$ with the $Oxy$ plane.

Find the derivative of the function

at the point $M(1,2,-2)$ along the direction of the tangent line, at this point, to the curve $x=t$, $y=2t^{2}$, $z=-2t^{4}$.

Find the equation of the tangent to the curve

Find the tangent to the preceding curve that is parallel to the plane $x+3y+2z=0$.

Find the cosines of the angles with the axes for the tangent to the curve

For the curve

find the points at which the tangent is parallel to the plane $yOz$ or $xOz$.

Find the tangent to the curve

at the point $(1,3,4)$.

Find the tangent to the curve

at the point $(-2,1,6)$.

Find the tangent to the curve

at the point $(m,m,2m^{2})$.

Find the cosines of the angles with the axes for the tangent to the curve

The same question (find the cosines of the angles with the axes for the tangent) as in the preceding problem, for the circle

A tangent is drawn to the curve

at the point where $x=\dfrac {p+q}{2}$. Find its length from the point of tangency to the plane $x=0$.

Find the angle between the tangent to the curve

and the vector drawn from the origin to the point of tangency.

Find the normal plane to the curve

At the point where $x=\dfrac {p}{2}$, find the normal plane to the curve

Prove that the normal planes of the curve

pass through the line

Find the derivative of the vector function $\mathbf{r}(t)$ and write the equation of the tangent at an arbitrary point of its hodograph, if:

Find the equation of the tangent to the curve:

Write the equation of the tangent to the curve

at $t = \pi / 2$. What angle does this tangent make with the axis $Oz$?

Find the equation of the tangent line and the normal plane of the curve $x = t^{4}, y = t^{3}, z = t^{2}$ at an arbitrary point of it.

Find the tangent to Viviani's curve (see problem 42) parallel to the plane $y = 0$.

At which points is the tangent to the curve $x = 3 t-t^{3}, y = 3 t^{2}, z = 3 t+t^{3}$ parallel to the plane $3 x+y+z+2 = 0$?

Find the normal plane of the curve $z = x^{2}+y^{2}, y = x$, perpendicular to the line $x = y = z$.

Find the tangent to the curve $x^{2}+y^{2} = 10, y^{2}+z^{2} = 25$ at the point $(1 ; 3 ; 4)$.

Find the cosines of the angles with the coordinate axes made by the tangents to the curve $x^{2} = 2 a z, y^{2} = 2 b z$.

A tangent is drawn to the curve $y^{2} = 2 p x, z^{2} = 2 q x$ at the point where $x = (p+q) / 2$. Find the length of the segment of this tangent from the point of tangency to the plane $x = 0$.

Prove that the normal planes of the curve $x = a \cos t, y = a \sin \alpha \sin t, z = a \cos \alpha \sin t$ pass through the line $x = 0, z + y \operatorname {tg} \alpha = 0$.

Prove that the tangents to the curve $x^{2} = 3 y, 2 x y = 9 z$ make a constant angle with some fixed direction.

The coordinates of the points of a certain curve satisfy the relation

Prove that the tangents to this curve are tangent to the sphere $x^{2}+y^{2} + z^{2} = a^{2}$.

Prove that the tangents to the curve

intersect the plane of the variables $x$ and $y$ in the circle $x^{2}+y^{2} = 4 a^{2}$.

Write the equation of the tangent and the normal plane of the curve $x = t, y = t^{2}, z = t^{3}$ at the point $(1 ; 1 ; 1)$. What curve results from the intersection of the tangents with the plane of the variables $x, y$?

Find the equation of the normal plane at an arbitrary point of the curve $x^{2}+y^{2} = 1, y^{2}+z^{2} = 1(y \neq \pm 1)$.

Prove that all normal planes of Viviani's curve $x = a \sin^{2} t, y = a \sin t \cos t, z = a \cos t$ pass through the origin.

Prove that if all normal planes of a space curve pass through a fixed point, then the curve is spherical.

Prove that the curve $x = e^{t} \cos t, y = e^{t} \sin t, z = e^{t}$ intersects all the generators of the cone $x^{2}+y^{2} = z^{2}$ at one and the same angle.

Prove that the curves of intersection of the cylinders $y^{2}+z^{2} = b^{2}$ with the surface $xy = az$ intersect, at a right angle, all the generators of this surface belonging to one family.

Prove that the curve $x = a \operatorname {tg} t, y = b \cos t, z = b \sin t$ lies on the surface of a paraboloid and intersects, at a right angle, all its generators of one family.

The curve called a loxodrome is defined by the equation

where $\theta$ is the latitude and $\varphi$ the longitude of a point on the sphere. Prove that it intersects the meridians of the sphere at an angle $\alpha$ whose tangent equals $a$.

Fig. 24.8

Let us formulate the definition of the stereographic projection of a plane onto a sphere tangent to it. Suppose that in the space $\mathbb {R}^{3}$ a Cartesian rectangular coordinate system $x, y, z$ is fixed, and a sphere of radius $a / 2$ with center at the point $B = (0 ; 0 ; a / 2)$, tangent to the plane of the variables $x$ and $y$ (Fig. 24.8), is given. Its equation has the form

Let us join by a line the top point of the sphere, i.e., the point $A = (0 ; 0 ; a)$, with an arbitrarily fixed point $M = (x ; y ; 0)$ of the plane of the variables $x$ and $y$. Then the point $M_{1} = \left(x_{1} ; y_{1} ; z_{1}\right)$ at which this line intersects the sphere (53) is called the stereographic projection of the point $M$ onto the sphere (53), and the point $M$ is called the stereographic projection of the point $M_{1}$ of the sphere onto the plane in question. The stereographic projection establishes a one-to-one correspondence between the points of the plane and the sphere with the point $A$ removed. The coordinates of the points $M$ and $M_{1}$ are related by

Write the equations of the osculating, normal, and rectifying planes at an arbitrary point of the curve: