Pedal Equations and Derivative of an Arc
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Derivative of Arc
1. Cartesian Equation
\[\frac{ds}{dx}=\sqrt{1+{{\left( \frac{dy}{dx} \right)}^{2}}}\]and \[\frac{ds}{dy}=\sqrt{1+{{\left( \frac{dx}{dy} \right)}^{2}}}\]
2. Polar Equation
\[\frac{ds}{dr}=\sqrt{1+{{\left( r\frac{d\theta }{dr} \right)}^{2}}}\]and \[\frac{ds}{d\theta }=\sqrt{{{r}^{2}}+{{\left( \frac{dr}{d\theta } \right)}^{2}}}\]
3. Parametric equation
\[\frac{ds}{dt}=\sqrt{{{\left( \frac{dx}{dt} \right)}^{2}}+{{\left( \frac{dy}{dt} \right)}^{2}}}\]
Co-ordinate
System
1. Cartesian
Co-ordinate
2. Polar
Co-ordinate
Relation
between Cartesian and Polar Co-ordinate
\[x=r\cos
\theta \]\[y=r\sin \theta \]\[{{r}^{2}}={{x}^{2}}+{{y}^{2}}\] \[\theta ={{\tan }^{-1}}\left( \frac{y}{x} \right)\]
Angle
between radius vector and tangent
\[\tan
\phi =r\frac{d\theta }{dr}\]
Length of perpendicular from pole to the tangent
1. In Polar Form \[p=r\sin \phi \] \[\frac{1}{{{p}^{2}}}=\frac{1}{{{r}^{2}}}+\frac{1}{{{r}^{4}}}{{\left( \frac{dr}{d\theta } \right)}^{2}}\]
2. In Cartesian form \[p=\left| \frac{x\frac{dy}{dx}-y}{1+{{\left( \frac{dy}{dx} \right)}^{2}}} \right|\]
Angle
between two curves
1. Cartesian
form
\[\tan
\psi =\frac{\tan {{\psi }_{1}}-\tan {{\psi }_{2}}}{1+\tan {{\psi }_{1}}\tan
{{\psi }_{2}}}\] \[\tan \psi =\frac{dy}{dx}\]
2. Polar
form
\[\tan
\phi =\frac{\tan {{\phi }_{1}}-\tan {{\phi }_{2}}}{1+\tan {{\phi }_{1}}\tan
{{\phi }_{2}}}\] \[\tan \phi =r\frac{d\theta }{dr}\]
Length of
Polar tangent, Normal, subtangent and subnormal
1.
\[\text{Length of Polar tangent}=r\sqrt{1+{{\left(
r\frac{d\theta }{dr} \right)}^{2}}}\]
2.
\[\text{Length of Polar
subtangent}={{r}^{2}}\frac{d\theta }{dr}\]
3.
\[\text{Length of Polar normal}=r\sqrt{1+{{\left(
\frac{1}{r}\frac{dr}{d\theta } \right)}^{2}}}\]
Pedal Equation
Pedal
Equation: Equation in p and r of the curve is said to be pedal equation
of curve.
1. Form Cartesian
curve (using 3 equations)
\[f\left(
x,y \right)=0\text{
}...\left( 1 \right)\]
\[{{r}^{2}}={{x}^{2}}+{{y}^{2}}\text{ }...\left( 2 \right)\]
\[p=\left|
\frac{x\frac{dy}{dx}-y}{1+{{\left( \frac{dy}{dx} \right)}^{2}}}
\right|\text{ }...\left( 3
\right)\]
2. Form
polar curve (using 2 equations)
\[f\left(
r,\theta \right)=0\text{ }...\left( 1 \right)\]
\[\frac{1}{{{p}^{2}}}=\frac{1}{{{r}^{2}}}+\frac{1}{{{r}^{4}}}{{\left(
\frac{dr}{d\theta } \right)}^{2}}\text{ }...\left( 2 \right)\]
3. Parametric
curves (using 4 equations)
\[x={{f}_{1}}\left(
t \right)\text{ }...\left(
1 \right)\]
\[x={{f}_{2}}\left(
t \right)\text{ }...\left(
2 \right)\]
\[{{r}^{2}}={{x}^{2}}+{{y}^{2}}\text{ }...\left( 3 \right)\]
\[p=\left|
\frac{x\frac{dy}{dx}-y}{1+{{\left( \frac{dy}{dx} \right)}^{2}}}
\right|\text{ }...\left( 4 \right)\]
Related Problem
Q.1 With
usual notation, prove that
\[tan
\varphi =\frac{x\frac{dy}{dx}-y}{x+y\frac{dy}{dx}}\]
Q.2 Find the
angle between the radius vector and the tangent at and point of curve \[r=a(1-\cos \theta )\text{ }\]
Q.3 Prove that for the curve $r=a{{e}^{\theta \cot \alpha }}$ the tangent is inclined at a constant angle with the radius vector.
Q.3 Prove that for the curve $r=a{{e}^{\theta \cot \alpha }}$ the tangent is inclined at a constant angle with the radius vector.
Q.4 Find the
angle of intersection of the following cardioids:
\[r=a(1+\cos \theta )\]
and
\[r=b(1-\cos \theta )\]
Q.5 In the
ellipse $\frac{l}{r}=1+e\cos \theta $, find the length of
a)
Polar sub tangent
b)
The perpendicular form the pole on the tangent.
Q.6 Prove
that the locus of the extremity of the polar subnormal of the curve $r=f(\theta
)\text{ }$is the curve $r=f'(\theta -{\pi }/{2}\;)\text{ }$
Q.7 Find the
pedal equation of the Cardioid $r=a(1-\cos \theta )\text{ }$
Q.8 Find the
pedal equation of the parabola ${{y}^{2}}=4a(x+a)\text{ }$
Q.9 Show that
the pedal equation of the ellipse $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$
is $\frac{1}{{{p}^{2}}}=\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}-\frac{{{r}^{2}}}{{{a}^{2}}{{b}^{2}}}$
Q.10
Find the pedal equation of the following Astroid:
$x=a{{\cos }^{3}}t$ , $y=a{{\sin
}^{3}}t$ or ${{x}^{2/3}}+{{y}^{2/3}}={{a}^{2/3}}$
Q.11
Find the angle $\varphi $ for the following curves
a)
$r=a{{e}^{b\theta }}$
b)
${{r}^{m}}={{a}^{m}}\cos m\theta \text{ }$
Q.12
Find the
angle of intersection of the following curves
a)
$r=a\theta $
and $r\theta =a$
b)
$r=2\sin \theta $ and
$r=2\cos \theta $
c)
$r=a\sin 2\theta $ and
$r=a\cos 2\theta $
d)
$r=\sin \theta +\cos \theta $ and
$r=2\sin \theta $
e)
${{r}^{2}}=16\sin 2\theta $ and
${{r}^{2}}\sin 2\theta =4$
Q.13
Prove that the following curves intersect
orthogonally
a)
$r=a\sin \theta $ and
$r=a\cos \theta $
b) $r=a\left( 1-\cos \theta \right)$
and $r=a\left( 1+\cos
\theta \right)$
c) $r=a\left( 1-\sin \theta \right)$
and $r=a\left( 1+\sin
\theta \right)$
d)
$\frac{a}{r}=1+\cos \theta $ and
$\frac{b}{r}=1-\cos \theta $
e)
${{r}^{m}}={{a}^{m}}\cos m\theta \text{ }$ and
${{r}^{m}}={{b}^{m}}\sin m\theta \text{ }$
Q.14
Find the length of polar subtangent for the
following curves
a)
$r=a(1-\cos \theta )\text{ }$
b)
$r=a(1+\cos \theta )\text{ }$
Q.15
Find the length of polar subnormal for the
following curves
a)
$r=a(1+\cos \theta )\text{ }$
b)
${{r}^{2}}={{a}^{2}}\cos 2\theta \text{ }$
Q.16
Show that for the curve $r=a\theta $, polar
subnormal is constant.
Q.17
Show that for the curve $r\theta =a$, polar
subtangent is constant.
Q.18
Show that the length of the polar tangent is
constant for the cure
$\theta ={{\cos
}^{-1}}\left( {r}/{a}\; \right)-\left( {1}/{r}\;
\right)\sqrt{a{}^{2}-{{r}^{2}}}$
Q.19
Show that the locus of the extremity of the polar
subtangent of the curve $\frac{1}{r}=f(\theta )$ is $\frac{1}{r}+f\left( \theta
+\frac{\pi }{2} \right)=0$
Q.20
Find the length of the perpendicular from the pole
on a tangent to the curve $r(\theta -1)=a{{\theta }^{2}}$
Q.21
Find the pedal equation of the curve $r=a(1+\cos
\theta )\text{ }$
Q.22
Find the pedal equation of the curve ${{r}^{2}}={{a}^{2}}\cos
2\theta \text{ }$
Q.23
Find the pedal equation of the curve ${{r}^{2}}\cos
2\theta \text{ }={{a}^{2}}$
Q.24
Find the pedal equation of the curve $r=a\operatorname{sech}(n\theta
)\text{ }$
Q.25
Find the pedal equation of the curve ${{r}^{m}}={{a}^{m}}\cos
m\theta \text{ }$
Q.26
Find the pedal equation of the curve ${{r}^{n}}={{a}^{n}}\sin
n\theta \text{ }$
Q.27
Find the pedal equation of the curve $r\theta
=a\text{ }$
Q.28
Find the pedal equation of the curve $r=a\theta
\text{ }$
Q.29
Find the pedal equation of the curve ${{x}^{\text{2}}}+{{y}^{2}}=2ax$
Q.30
Find the pedal equation of the curve ${{x}^{\text{2}}}-{{y}^{2}}={{a}^{2}}$
Q.31
Find the pedal equation of the curve $\frac{{{x}^{\text{2}}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$
Q.32
For the parabola ${{y}^{2}}=4ax$, prove that \[\frac{ds}{dx}=\sqrt{\frac{a+x}{x}}\]
Q.33
For the curve $x=a(1-\cos t)\text{ }$$y=a(t+\sin
t)\text{ }$, find
Q.34
Find $\frac{ds}{d\theta }$ for the curve $r=a(1+\cos
\theta )\text{ }$
Q.35
Find $\frac{ds}{d\theta }$ for the curve $\frac{2a}{r}=1+\cos
\theta $
Q.36
Find $\frac{ds}{dt}$ for the curve \[x\sin
t+y\cos t=f'(t)\] \[x\cos t-y\sin
t=f''(t)\]
Q.37 Prove \[\frac{ds}{d\theta
}=\frac{{{r}^{2}}}{p}\] and \[\frac{ds}{dr}=\frac{r}{\sqrt{{{r}^{2}}-{{p}^{2}}}}\]
Q.38
For the curve ${{r}^{m}}={{a}^{m}}\cos m\theta
\text{ }$, prove that
a)
\[\frac{ds}{d\theta }=\frac{{{a}^{m}}}{{{r}^{m-1}}}\text{=}a{{\left(
\sec m\theta
\right)}^{{}^{m-1}/{}_{m}}}\text{ }\]
b)
\[{{a}^{2m}}\frac{{{d}^{2}}r}{d{{s}^{2}}}+m{{r}^{2m-1}}=0\]
Q.39
For the curve \[{{y}^{2}}={{c}^{2}}+{{s}^{2}}\]
prove that \[\frac{dy}{dx}=\frac{\sqrt{{{y}^{2}}-{{c}^{2}}}}{c}\] Hence show
that perpendicular from the foot of the ordinate upon the tangent is of
constant length.
Q.40
If $r=a{{e}^{\theta \cot \alpha }}$ then prove
that
\[(a)\text{
}s=cr\]\[(b)\text{ }\frac{dr}{ds}=\cos \alpha \]
Q.41
For any curve prove that \[{{\sin }^{2}}\phi
\frac{d\phi }{d\theta }+r\frac{{{d}^{2}}r}{d{{s}^{2}}}=0\]
Q.42
If $\frac{2a}{r}=1+\cos \theta $ then with usual
notation show that \[\frac{ds}{d\psi }=\frac{2a}{{{\sin }^{3}}\psi }\]
Q.43 For the parabola $\frac{2a}{r}=1-\cos \theta $, prove that
a) $p=a\cos ec\left( {\theta }/{2}\; \right)$
b) $\text{Polar subtangent}=\text{ }2a\cos ec\theta $
c) Pedal equation \({p}^{2}=ar\)
