Tag: Incomplete

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

by RoRi

May 31, 2016

We that points $P,Q$ are symmetric with respect to a circle if $P$ and $Q$ 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 $Q$ describes the straight line $x + 2y - 5 = 0$ , find the set of points $P$ symmetric to $Q$ with respect to the circle $x^2 + y^2 = 4$ .

Incomplete.

Prove some properties of parabolas

by RoRi

May 31, 2016

A chord with length $8 |c|$ is perpendicular to the axis of the parabola $y^2 = 4cx$ . If $P$ and $Q$ are the points where the chord and the parabola meet, prove that the vector from $O$ to $P$ is perpendicular to the vector from $O$ to $Q$ .
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

by RoRi

May 31, 2016

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

by RoRi

May 31, 2016

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

by RoRi

May 31, 2016

Consider the Cartesian equation
$\frac{x^2}{a^2} + \frac{y^2}{a^2 - c^2} = 1.$

Prove that this equation represents all conic sections symmetric about the origin with foci at $(c,0)$ and $(-c,0)$ .
Let $c$ be a fixed constant and let $S$ be the set of all such conics as $a^2$ takes on all positive values other than $c^2$ . Prove that every curve in the set $S$ satisfies the differential equation
$xy \left( \frac{dx}{dy} \right)^2 + (x^2 - y^2 - c^2) \frac{dy}{dx} - xy = 0.$
Prove that the set $S$ is self-orthogonal. This means that the set of all orthogonal trajectories of curve in $S$ is the set $S$ itself.

Incomplete.

Prove that two parabolas intersect orthogonally

by RoRi

May 31, 2016

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

by RoRi

May 31, 2016

Consider the set of points $P$ such that the distance from $P$ to the point $(2,3)$ is equal to the sum of the distances from $P$ to the two coordinate axes.

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.
Sketch the set of points in the other quadrants.

Incomplete.

Find condition on constants such that two parabolas are tangent

by RoRi

May 31, 2016

Let $a \neq 0$ , and consider the two parabolas
$y^2 = 4p(x-a), \qquad \text{and} \qquad x^2 = 4qy.$

If these two parabolas are tangent to each other, show that the $x$ -coordinate of the contact point depends only on $a$ .
Find conditions for the constants $a,p,q$ such that the two parabolas are tangent to each other.