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Exercises and solutions from textbooks

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

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.


Prove some properties of parabolas

  1. 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.
  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.


Prove some properties of conic sections

  1. 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).

  2. 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

        \[ xy \left( \frac{dx}{dy} \right)^2 + (x^2 - y^2 - c^2) \frac{dy}{dx} - xy = 0. \]

  3. 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.