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# Find the set of points symmetric to a given point with respect to a circle

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.

# Prove some properties of parabolas

1. A chord with length is perpendicular to the axis of the parabola . If and are the points where the chord and the parabola meet, prove that the vector from to is perpendicular to the vector from to .
2. Show that the length of the chord of a parabola drawn through the focus and parallel to the directrix (the latus rectum) is twice the distance from the focus to the directrix. Next, show that the tangents to the parabola at both ends of this chord intersect the axis of the parabola on the directrix.

Incomplete.

# Show that the centers of a family of circles form a parabola

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.

# Show that the centers of the family of circles passing through a point is a parabola

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 some properties of conic sections

1. Consider the Cartesian equation

Prove that this equation represents all conic sections symmetric about the origin with foci at and .

2. Let be a fixed constant and let be the set of all such conics as takes on all positive values other than . Prove that every curve in the set satisfies the differential equation

3. Prove that the set is self-orthogonal. This means that the set of all orthogonal trajectories of curve in is the set itself.

Incomplete.

# Prove that two parabolas intersect orthogonally

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.

# Show that a locus of points is a hyperbola

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.

1. Show that the part of this set of points lying in the first quadrant forms a hyperbola. Locate the asymptotes of this hyperbola and make a sketch.
2. Sketch the set of points in the other quadrants.

Incomplete.

# Find condition on constants such that two parabolas are tangent

1. Let , and consider the two parabolas

If these two parabolas are tangent to each other, show that the -coordinate of the contact point depends only on .

2. Find conditions for the constants such that the two parabolas are tangent to each other.

Incomplete.

# Find the point of contact between a parabola and a line tangent to the parabola

The line is tangent to the parabola with equation . Find the point at which the line touches the parabola.

Incomplete.

# Prove that the collection of all parabolas is invariant under similarity transforms

1. Prove that the collection of all parabolas is invariant under similarity transforms.
2. Find all parabolas similar to the parabola .

Incomplete.