Mathematical Immaturity

13.25 Miscellaneous exercises on conic sections

1. Show that the area of the region bounded by the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ is $ab$ times the area of a circle with radius 1. Note: This statement can be proved from general properties of the integral, without performing any integrations.
2. (a) Show that the volume of the solid of revolution generated by rotating the ellipse $$ \begin{align*} \\ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \end{align*} $$ about its major axis is $ab^{2}$ times the volume of a unit sphere. Note: This statement can be proved from general properties of the integral, without performing any integrations. (b) What is the result if it is rotated about its minor axis?
3. Find all positive numbers $A$ and $B,$ $A\ > B,$ such that the area of the region enclosed by the ellipse $Ax^{2} + By^{2} = 3$ is equal to the area of the region enclosed by the ellipse $$ \begin{align*} \\ (A + B)x^{2} + (A - B)y^{2} = 3 \end{align*} $$
4. A parabolic arch has base of length $b$ and height $h.$ Determine the area of the region bounded by the arch and the base.
5. The region bounded by the parabola $y^{2} = 8x$ and the line $x = 2$ is rotated about the $x$ axis. Find the volume of the solid of revolution so generated.
6. Two parabolas having the equations $y^{2} = 2(x - 1)$ and $y^{2} = 4(x - 2)$ enclose a plane region $R.$ (a) Compute the area of $R$ by integration. (b) Find the volume of the solid of revolution generated by revolving $R$ about the $x$-axis. (c) Same as (b), but revolve $R$ about the $y$-axis.
7. Find a Cartesian equation for the conic section consisting of all points $(x, y)$ whose distance from the point $(0, 2)$ is half the distance from the line $y=8$
8. Find a Cartesian equation for the parabola whose focus is at the origin and whose directrix is the line $x + y + 1 = 0$
9. Find a Cartesian equation for a hyperbola passing through the origin, given that its asymptotes are the lines $y = 2x + 1$ and $y = -2x + 3$
10. (a) For each $p>0,$ the equation $px^{2} + (p+2)y^{2} = p^{2} + 2p$ represents an ellipse. Find (in terms of $p$) the eccentricity and the coordinates of the foci. (b) Find a Cartesian equation for the hyperbola which has the same foci as the ellipse of part (a) and which has eccentricity $\sqrt{3}.$
11. In section 13.22 we proved that a conic symmetric about the origin satisfies the equation $\|X - F\| = \left|eX \cdot N - a\right|,$ where $a = ed + eF\cdot N.$ Use this relation to prove that $\|X - F\| + \|X + F\| = 2a$ if the conic is an ellipse. In other words, the sum of the distances from any point on an ellipse to its foci is constant.
12. Refer to Exercise 11. Prove that on each branch of a hyperbola the difference $\|X - F\|$ - $\|X + F\|$ is constant.
13. (a) Prove that a similarity transformation (replacing $x$ by $tx$ and $y$ by $ty$) carries an ellipse with center at the origin into another ellipse with the same eccentricity. In other words, similar ellipses have the same eccentricity. (b) Prove also the converse. That is, if two concentric ellipses have the same eccentricity and major axes on the same line, then they are related by similarity transform. (c) Prove results corresponding to (a) and (b) for hyperbolas.
14. Use the Cartesian equation which represents all conics of eccentricity $e$ and center at the origin to prove that these conics are integral curves of the differential equation $y' = (e^{2} - 1)x/y.$ Note: Since this is a homogeneous differential equation (Section 8.25), the set of all such conics of eccentricity $e$ is invariant under a similarity transformation. (Compare with exercise 13.)
15. (a) Prove that the collection of all parabolas is invariant under a similarity transformation. That is, a similarity transformation carries a parabola into a parabola. (b) Find all the parabolas similar to $y = x^2.$
16. The line $x - y + 4 = 0$ is tangent to the parabola $y^2 = 16x.$ Find the point of contact.
17. (a) Given $a ≠ 0.$ If the two parabolas $y^2 = 4p(x - a)$ and $x^2 = 4qy$ are tangent to each other, show that the $x$-coordinate of the point of contact is determined by $a$ alone. (b) Find a condition on $a,$ $p,$ and $q$ which expresses the fact that the two parabolas are tangent to each other.
18. Consider the locus of the points $P$ in the plane for which the distance of $P$ from the point $(2, 3)$ is equal to the sum of the distances of $P$ from the two coordinate axes. (a) Show that the part of this locus which lies in the first quadrant is part of a hyperbola. Locate the asymptotes and make a sketch. (b) Sketch the graph of the locus in the other quadrants.
19. Two parabolas have the same point as focus and the same line as axis, but their vertices lie on opposite sides of the focus. Prove that the parabolas intersect orthogonally (i.e., their tangent lines are perpendicular at the points of intersection).
20. (a) Prove that the Cartesian equation $$ \begin{align*} \\ \frac{x^2}{a^2} + \frac{y^2}{a^2 - c^2} = 1 \end{align*} $$ represents all conics symmetric about the origin with foci at $(c, 0)$ and $(-c, 0).$ (b) Keep $c$ fixed and let $S$ denote the set of all such conics obtained as $a^2$ varies over all positive numbers $> c^2.$ Prove that every curve in $S$ satisfies the differential equation $$ \begin{align*} \\ xy\left(\frac{dy}{dx}\right)^2 + (x^2 - y^2 - c^2)\frac{dy}{dx} - xy = 0. \end{align*} $$ (c) Prove that $S$ is self-orthogonal; that is, the set of all orthogonal trajectories of curves in $S$ is $S$ itself. [Hint: Replace $y'$ by $-1/y'$ in the differential equation in (b).]
21. Show that the locus of the centers of a family of circles, all of which pass through a given point and are tangent to a given line, is a parabola.
22. Show that the locus of the centers of a family of circles, all of which are tangent (externally) to a given circle and also to a given straight line, is a parabola. (Exercise 21 can be considered to be a special case.)
23. (a) A chord of length $8|c|$ is drawn perpendicular to the axis of the parabola $y^2 = 4cx.$ Let $P$ and $Q$ be the points where the chord meets the parabola. Show that the vector from $O$ to $P$ is perpendicular to that from $O$ to $Q.$ (b) The chord of a parabola drawn through the focus and parallel to the directrix is called the latus rectum. Show first that the length of the latus rectum is twice the distance from the focus to the directrix, and then show that the tangents to the parabola at both ends of the latus rectum intersect the axis of the parabola on the directrix.
24. Two points $P$ and $Q$ are said to be symmetric with respect to a circle if $P$ and $Q$ are collinear with the center, if the center is not between them, and if the product of their distances from the center is equal to the square of the radius. Given that $Q$ describes the straight line $x + 2y - 5 = 0,$ find the locus of the point $P$ symmetric to $Q$ with respect to the circle $x^2 + y^2 = 4.$