Exercises

1. Find the equations of tangent and normal, lengths of subtangent, subnormal tangent, and normal at the point $(a,a)$ on the cissoid $y^2 = \frac {x^3}{2a - x}$.

   Figure 6.5: Graph of cissoid $y^2 = \frac {x^3}{2a - x}$ with $a=1$.

   

   Solution. $\frac{dy}{dx} =\ \frac{3ax^2 - x^3}{y(2a - x)^2}$. Hence

   $$\frac{dy_1}{dx_1} = \left [ \frac{dy}{dx} \right ]_{x = a, y = a} =\ \frac{3a^3 - a^3}{a(2a - a)^2} = 2$$
   is the slope of tangent. Substituting in (6.1) gives
   $$y = 2x - a,$$
   the equation of the tangent line. Substituting in (6.2) gives
   $$2y + x = 3a,$$
   the equation of the normal line. Substituting in (6.3) gives
   $$TM = \frac{a}{2},$$
   the length of subtangent. Substituting in (6.4) gives
   $$MN = 2a,$$
   the length of subnormal. Also
   $$PT = \sqrt{(TM)^2 + (MP)^2} = \sqrt{\frac{a^2}{4} + a^2} = \frac{a}{2} \sqrt{5},$$
   which is the length of tangent, and
   $$PN = \sqrt{(MN)^2 + (MP)^2} = \sqrt{4a^2 + a^2} = a \sqrt{5},$$
   the length of normal.

2. Find equations of tangent and normal to the ellipse $x^2 + 2y^2 - 2xy - x = 0$ at the points where $x=1$.

   Ans. At $(1,0)$, $2y = x - 1$, $y + 2x = 2$. At $(1, 1)$, $2y = x + 1$, $y + 2x = 3$.

3. Find equations of tangent and normal, lengths of subtangent and subnormal at the point $(x_1,y_1)$ on the circle^6.5 $x^2 + y^2 = r^2$.

   Ans. $x_lx + y_1y = r^2$, $x_1y - y_1x = 0$, $-x_1, -\frac{{y_1}^2}{x_1}$.

4. Show that the subtangent to the parabola $y^2 = 4px$ is bisected at the vertex, and that the subnormal is constant and equal to $2p$.

5. Find the equation of the tangent at $(x_1,y_1)$ to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$.

   Ans. $\frac{x_1x}{a^2} + \frac{y_1y}{b^2} = 1$.

   Here's how to find the length of tangent, normal, subtangent and subnormal of this in SAGE using the values $a=1$, $b=2$ (so $x^2 + \frac{y^2}{4} = 1$) and $x_1=4/5$, $y_1=6/5$.

   [fontsize=\small,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
   
   sage: x = var("x")
   sage: y = var("y")
   sage: F = x^2 + y^2/4 - 1
   sage: Dx = -diff(F,y)/diff(F,x); Dx; Dx(4/5,6/5)
   -y/(4*x)
   -3/8
   sage: Dy = -diff(F,x)/diff(F,y); Dy; Dy(4/5,6/5)
   -4*x/y
   -8/3
   

   (For this SAGE calculation, we have used the fact that $F(x,y)=0$ implies $F_x(x,y)+\frac{dy}{dx}F_y(x,y)=0$, where $y$ is regarded as a function of $x$.) Therefore, we have (using (6.3))

   $${\rm length\ of\ subtangent} = y_1 \frac{dx}{dy}\vert _{x=x_1,y=y_1} = (6/5)(-3/8)=-9/20,$$
   (using (6.4))
   $${\rm\ length\ of\ subnormal} = y_1 \frac{dy}{dx}\vert _{x=x_1,y=y_1} = (6/5)(-8/3)=-16/5,$$
   (using (6.5))
   $$\begin{array}{ll} {\rm length\ of\ tangent} =y_1 \sqrt{ \lef... ...rt{1+\frac{9}{64}}\\ &=3\sqrt{73}/20 =1.2816...\ , \end{array}$$
   and (using (6.6))
   $$\begin{array}{ll} {\rm length\ of\ normal} = y_1 \sqrt{ \lef... ...qrt{1+\frac{64}{9}}\\ &=2\sqrt{73}/5 =3.4176...\ . \end{array}$$

6. Find equations of tangent and normal to the Witch of Agnesi $y = \frac{8a^3}{4a^2 + x^2}$ as at the point where $x = 2a$.

   Ans. $x + 2y = 4a$, $y = 2x - 3a$.

7. Prove that at any point on the catenary $y = \frac{a}{2}(e^{\frac{x}{a}} + e^{-\frac{x}{a}})$ the lengths of subnormal and normal are $\frac{a}{4}(e^{\frac{2x}{a}} - e^{-\frac{2x}{a}})$ and $\frac{y^2}{a}$ respectively.

8. Find equations of tangent and normal, lengths of subtangent and subnormal, to each of the following curves at the points indicated:
   $$\begin{array}{ll} (a)\ \ y = x^3\ {\rm at}\ (\frac{1}{2}, \... ... (-3,-4) & (h)\ \ 2x^2 - y^2 = 14\ {\rm at}\ (3,-2) \end{array}$$

9. Prove that the length of subtangent to $y = a^x$ is constant and equal to $\frac{1}{\log a}$.

10. Get the equation of tangent to the parabola $y^2 = 20x$ which makes an angle of $45^o=\pi/4$ with the $x$-axis.

    Ans. $y = x + 5$. (Hint: First find point of contact by method of Example 6.1.1.)

11. Find equations of tangents to the circle $x^2 + y^2 = 52$ which are parallel to the line $2x + 3y = 6$.

    Ans. $2x + 3y \pm 26 = 0$

12. Find equations of tangents to the hyperbola $4x^2 - 9y^2 + 36 = 0$ which are perpendicular to the line $2y + 5x = 10$.

    Ans. $2x - 5y \pm 8 = 0$.

13. Show that in the equilateral hyperbola $2xy = a^2$ the area of the triangle formed by a tangent and the coordinate axes is constant and equal to $a^2$.

14. Find equations of tangents and normals to the curve $y^2 = 2x^2 - x^3$ at the points where $x=1$.

    Ans. At $(1, 1)$, $2y = x + 1$, $y + 2x = 3$. At $(1,-1)$, $2y =-x-1$, $y-2x = -3$.

15. Show that the sum of the intercepts of the tangent to the parabola $x^{\frac{1}{2}} + y^{\frac{1}{2}} = a^{\frac{1}{2}}$.

16. Find the equation of tangent to the curve $x^2(x + y) = a^2(x-y)$ at the origin.

17. Show that for the hypocycloid $x^{\frac{2}{3}} + y^{\frac{2}{3}} = a^{\frac{2}{3}}$ that portion of the tangent included between the coordinate axes is constant and equal to $a$.

    (This curve is parameterized by $x=a\cos(t)^3$, $y=a\sin(t)^3$, $0\leq t\leq 2\pi$. Parametric equations shall be discussed in the next section.)

18. Show that the curve $y = ae^{\frac{x}{c}}$ has a constant subtangent.