Compute the derivatives and of the vector valued function
We compute,
Compute the derivatives and of the vector valued function
We compute,
We that points are symmetric with respect to a circle if and are on a line with the center of the circle, the center is not between the points, and the product of their distances from the center is equal to the square of the radius of the circle. Given that describes the straight line , find the set of points symmetric to with respect to the circle .
Incomplete.
Incomplete.
Show that the centers of the family of circles all of which are tangent to a given circle and also to a given line form a parabola.
Proof.Incomplete.
Prove that the set of the centers of the family of circles all of which pass through a given point and are tangent to a given line forms a parabola.
Proof. Incomplete.
Prove that this equation represents all conic sections symmetric about the origin with foci at and .
Incomplete.
Consider two parabolas which have the same focus and the same line as axis, and let these two parabolas have vertices lying on opposite sides of the focus. Prove that the parabolas intersect orthogonally (i.e., their tangent lines are perpendicular at the point of intersection).
Proof.Incomplete.
Consider the set of points such that the distance from to the point is equal to the sum of the distances from to the two coordinate axes.
Incomplete.
If these two parabolas are tangent to each other, show that the -coordinate of the contact point depends only on .
Incomplete.
The line is tangent to the parabola with equation . Find the point at which the line touches the parabola.
Incomplete.