Exercises

Find equations of tangent and normal, lengths of subtangent and subnormal to each of the following curves at the point indicated:

1. Curve: $x = t^2$, $2y = t$;

   Point: $t = 1$.

   Tangent line: $x - 4y + 1 = 0$;

   Normal line: $8x + 2y - 9 = 0$;

   Subtangent: $2$;

   Subnormal: $\frac{1}{8}$.

2. Curve: $x = t$, $y = t^3$;

   Point: $t = 2$.

   Tangent line: $12x - y - 16 = 0$;

   Normal line: $x + 12y - 98 = 0$;

   Subtangent: $\frac{2}{3}$;

   Subnormal: $96$.

3. Curve: $x = t^2$, $y = t^3$;

   Point: $t = 1$.

   Tangent line: $3x - 2y - 1 = 0$;

   Normal line: $2x + 3y - 5 = 0$;

   Subtangent: $\frac{2}{3}$;

   Subnormal: $\frac{3}{2}$.

4. Curve: $x = 2e^t$, $y = e^{- t}$;

   Point: $t = 0$.

   Tangent line: $x + 2y - 4 = 0$;

   Normal line: $2x - y - 3 = 0$;

   Subtangent: $-2$;

   Subnormal: $-\frac{1}{2}$.

5. Curve: $x = \sin\, t$, $y = \cos\, 2t$;

   Point: $t = \frac{\pi}{6}$.

   Tangent line: $2y + 4x - 3 = 0$;

   Normal line: $4y - 2x - 1 = 0$;

   Subtangent: $-\frac{1}{4}$;

   Subnormal: $-1$.

   SAGE can help with the computations here:

   [fontsize=\small,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
   
   sage: t = var("t")
   sage: x = sin(t)
   sage: y = cos(2*t)
   sage: t0 = pi/6
   sage: y_x = diff(y,t)/diff(x,t)
   sage: y_x
   -2*sin(2*t)/cos(t)
   sage: y_x(t0)
   -2
   sage: m = y_x(t0); x0 = x(t0); y0 = y(t0)
   sage: X,Y = var("X,Y")
   sage: Y - y0 == m*(X - x0)
   Y - 1/2 == -2*(X - 1/2)
   

   The last line is the point-slope form of the tangent line of the parametric curve at that point $t_0=\pi/6$ (so, $(x_0,y_0)=(\sin(t_0),\cos(2t_0))=( 1/2, 1/2)$). We use $X$ and $Y$ in place of $x$ and $y$ so as to not over-ride the entries that SAGE has stored for them. Continuing the above SAGE computations:

   [fontsize=\small,fontfamily=courier,fontshape=tt,frame=single,label=\sage]
   
   sage: x_y = diff(x,t)/diff(y,t)
   sage: len_subtan = y(t0)*x_y(t0); len_subtan
   -1/4
   sage:
   sage: len_subnor = y(t0)*y_x(t0); len_subnor
   -1
   sage: len_tan = y(t0)*sqrt(x_y(t0)^2+1); len_tan
   sqrt(5)/4
   sage: len_nor = y(t0)*sqrt(y_x(t0)^2+1); len_nor
   sqrt(5)/2
   

   These tell us the length of the subtangent is $-\frac{1}{4}$ (as expected), as well as the lengths of the subnormal, tangent and normal, using formulas (6.10), (6.3), (6.4), (6.5), (6.6) of the last section.

6. Curve: $x = 1 - t$, $y = t^2$;

   Point: $t = 3$.

7. Curve: $x = 3t$; $y = 6t - t^2$;

   Point: $t = 0$.

8. Curve: $x = t^3$; $y = t$;

   Point: $t = 2$.

9. Curve: $x = t^3$, $y = t^2$;

   Point: $t = - 1$.

10. Curve: $x = 2 - t$; $y = 3t^2$;

    Point: $t = 1$.

11. Curve: $x = \cos\, t$, $y = \sin\, 2t$;

    Point: $t = \frac{\pi}{3}$.

12. Curve: $x = 3e^{-t}$, $y = 2e^t$;

    Point: $t = 0$.

13. Curve: $x = \sin\, t$, $y = 2 \cos\, t$;

    Point: $t = \frac{\pi}{4}$.

14. Curve: $x = 4 \cos\, t$, $y = 3 \sin\, t$;

    Point: $t = \frac{\pi}{2}$.

15. Curve:

    Point:

In the following curves find lengths of (a) subtangent, (b) subnormal, (c) tangent, (d) normal, at any point:

1. The curve
   $$\begin{cases} x = a(\cos t + t \sin t), \\ y = a(\sin t - t \cos t). \end{cases}$$

   Ans. (a) $y\cot\, t$, (b) $y\tan\, t$, (c) $\frac{y}{\sin\, t}$, (d) $\frac{y}{\cos\, t}$.

2. The hypocycloid (astroid)
   $$\begin{cases} x = 4 a \cos^3 t, \\ y = 4a \sin^3 t. \end{cases}$$

   Ans. (a) $- y\cot\, t$, (b) $- y\tan\, t$, (c) $\frac{y}{\sin\, t}$, (d) $\frac{y}{\cos\, t}$.

3. The circle
   $$\begin{cases}x = r \cos\, t, \\ y = r \sin\, t. \end{cases}$$

4. The cardioid
   $$\begin{cases} x = a(2 \cos\, t - \cos\, 2t), \\ y = a(2 \sin\, t - \sin\, 2t). \end{cases}$$

5. The folium
   $$\begin{cases} x = \frac{3t}{1 + t^3} \\ y = \frac{3t^2}{1 + t^3}. \end{cases}$$

6. The hyperbolic spiral
   $$\begin{cases} x = \frac{a}{t} \cos\, t \\ y = \frac{a}{t} \sin\, t \end{cases}$$