Exercises

The corresponding figure should be drawn in each of the following examples:

1. Find the slope of $y = \frac{x}{1 + x^2}$ at the origin.

   Ans. $1 = \tan\, \tau$.

2. What angle does the tangent to the curve $x^2y^2 = a^3(x + y)$ at the origin make with the $x$-axis?

   Ans. $\tau = 135^o =3\pi/4$.

3. What is the direction in which the point generating the graph of $y = 3x^2 - x$ tends to move at the instant when $x=1$?

   Ans. Parallel to a line whose slope is $5$.

4. Show that $\frac{dy}{dx}$ (or slope) is constant for a straight line.

5. Find the points where the curve $y = x^3 - 3x^2 - 9x + 5$ is parallel to the $x$-axis.

   Ans. $x=3$, $x = -1$.

6. At what point on $y^2 = 2x^3$ is the slope equal to $3$?

   Ans. $(2, 4)$.

7. At what points on the circle $x^2 + y^2 = r^2$ is the slope of the tangent line equal to $-\frac{3}{4}$?

   Ans. $\left ( \pm \frac{3r}{5}, \pm \frac{4r}{5} \right )$

8. Where will a point moving on the parabola $y = x^2 - 7x + 3$ be moving parallel to the line $y = 5x + 2$?

   Ans. $(6, -3)$.

9. Find the points where a particle moving on the circle $x^2 + y^2 = 169$ moves perpendicular to the line $5x + 12y = 60$.

   Ans. $(\pm 12, \mp 5)$.

10. Show that all the curves of the system $y = \log\, kx$ have the same slope; i.e. the slope is independent of $k$.

11. The path of the projectile from a mortar cannon lies on the parabola $y = 2x - x^2$; the unit is 1 mile, the $x$-axis being horizontal and the $y$-axis vertical, and the origin being the point of projection. Find the direction of motion of the projectile

    (a) at instant of projection;

    (b) when it strikes a vertical cliff $\frac{3}{2}$ miles distant.

    (c) Where will the path make an inclination of $45^o=\pi/4$ with the horizontal?

    (d) Where will the projectile travel horizontally?

    Ans. (a) $\arctan\, 2$; (b) $135^o=3\pi/4$; (c) $(\frac{1}{2}, \frac{3}{4})$; (d) $(1, 1)$.

12. If the cannon in the preceding example was situated on a hillside of inclination $45^o=\pi/4$, at what angle would a shot fired up strike the hillside?

    Ans. $45^o=\pi/4$.

13. At what angles does a road following the line $3y - 2x - 8 = 0$ intersect a railway track following the parabola $y^2 = 8x$?

    Ans. $\arctan \frac{1}{5}$, and $\arctan \frac{1}{8}$.

14. Find the angle of intersection between the parabola $y^2 = 6x$ and the circle $x^2 + y^2 = 16$.

    Ans. $\arctan \frac{5}{3} \sqrt{3}$.

15. Show that the hyperbola $x^2 - y^2 = 5$ and the ellipse $\frac{x^2}{18} + \frac{y^2}{8} = 1$ intersect at right angles.

16. Show that the circle $x^2 + y^2 = 8ax$ and the cissoid $y^2 = \frac {x^3}{2a - x}$

    (a) are perpendicular at the origin;

    (b) intersect at an angle of $45^o=\pi/4$ at two other points.

17. Find the angle of intersection of the parabola $x^2 = 4ay$ and the Witch of Agnesi, $y = \frac{8a^3}{x^2 + 4a^2}$.

    Ans. $\arctan\, 3 = 71^o33' = 1.249...$.

    For the interesting history of this curve, see for example
    http://en.wikipedia.org/wiki/Witch_of_Agnesi.

18. Show that the tangents to the Folium of Descartes, $x^3 + y^3 = 3axy$ at the points where it meets the parabola $y^2 = ax$ are parallel to the $y$-axis.

    For some history of this curve, see for example
    http://en.wikipedia.org/wiki/Folium_of_Descartes.

19. At how many points will a particle moving on the curve $y = x^3 - 2x^2 + x - 4$ be moving parallel to the $x$-axis? What are the points?

    Ans. Two; at $(1,-4)$ and $(\frac{1}{3}, -\frac{104}{27})$.

20. Find the angle at which the parabolas $y = 3x^2 - 1$ and $y = 2x^2 + 3$ intersect.

    Ans. $\arctan \frac{4}{97}$.

21. Find the relation between the coefficients of the conics $a_1x^2 + b_1y^2 = 1$ and $a_2x^2 + b_2y^2 = 1$ when they intersect at right angles.

    Ans. $\frac{1}{a_1} - \frac{1}{b_1} = \frac{1}{b_2} - \frac{1}{b_2}$.