Pedal Equations and Derivative of an Arc
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Co-ordinate System
Cartesian Co-ordinate
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
\[\tan \phi =r\frac{d\theta }{dr}\]
Angle between two curves
Cartesian form
\[\tan \psi =\frac{\tan {{\psi }_{1}}-\tan {{\psi }_{2}}}{1+\tan {{\psi }_{1}}\tan {{\psi }_{2}}}\], \[\tan \psi =\frac{dy}{dx}\]
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
\[\text{Length of Polar tangent}=r\sqrt{1+{{\left( r\frac{d\theta }{dr} \right)}^{2}}}\]
\[\text{Length of Polar subtangent}={{r}^{2}}\frac{d\theta }{dr}\]
\[\text{Length of Polar normal}=r\sqrt{1+{{\left( \frac{1}{r}\frac{dr}{d\theta } \right)}^{2}}}\]
\[\text{Length of Polar subnormal}=\frac{dr}{d\theta }\]
Length of perpendicular from pole to the tangent p=rsinɸ 1/p^2 =1/r^2 +1/r^4 (dr/dθ)^2
\[p=r\sin \phi \], \[\frac{1}{{{p}^{2}}}=\frac{1}{{{r}^{2}}}+\frac{1}{{{r}^{4}}}{{\left( \frac{dr}{d\theta } \right)}^{2}}\]
Pedal Equation
For Cartesian curve
For polar curve
Derivative of Arc
Cartesian Equation
Polar Equation
Parametric equation
Related Problem
With usual notation, prove that
$\tan \varphi =\frac{x\frac{dy}{dx}-y}{x+y\frac{dy}{dx}}$
Find the angle between the radius vector and the tangent at and point of
following curves:
$r=a(1-\cos \theta )\text{ }\!\![\!\!\text{ cardioid }\!\!]\!\!\text{ }$
$\frac{2a}{r}=1+\cos \theta $
Find the angle of intersection of the following cardioids:
$r=a(1+\cos \theta )\text{ and }r=b(1-\cos \theta )$
In the ellipse $\frac{l}{r}=1+e\cos \theta $, find the length of
Polar sub tangent
The perpendicular form the pole on the tangent.
Prove that the locus of the extremity of the polar subnormal of the curve
$r=f(\theta )\text{ }$is the curve $r=f'(\theta -{\pi }/{2}\;)\text{ }$
For the parabola $\frac{2a}{r}=1-\cos \theta $, prove that
$p=a\cos ec\left( {\theta }/{2}\; \right)$
Polar subtangent = $2a\cos ec\left( \theta \right)$
Pedal equation ${{p}^{2}}=ar$
Find the pedal equation of the Cardioid $r=a(1-\cos \theta )\text{ }$
Find the pedal equation of the parabola ${{y}^{2}}=4a(x+a)\text{ }$
Show that the pedal equation of the ellipse $\frac{{{x}^{2}}}{{{a}^{2}}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1$ is
$\frac{1}{{{p}^{2}}}=\frac{1}{{{a}^{2}}}+\frac{1}{{{b}^{2}}}-\frac{{{r}^{2}}}{{{a}^{2}}{{b}^{2}}}$
Find the pedal equation of the following Astroid:
$x=a{{\cos }^{3}}t$ , $y=a{{\sin }^{3}}t$ or ${{x}^{2/3}}+{{y}^{2/3}}={{a}^{2/3}}$
Find the angle $\varphi $ for the following curves
$r=a{{e}^{b\theta }}$
${{r}^{m}}={{a}^{m}}\cos m\theta \text{ }$
Find the angle of intersection of the following curves
$r=a\theta $ and $r\theta =a$
$r=2\sin \theta $ and $r=2\cos \theta $
$r=a\sin 2\theta $ and $r=a\cos 2\theta $
$r=\sin \theta +\cos \theta $ and $r=2\sin \theta $
${{r}^{2}}=16\sin 2\theta $ and ${{r}^{2}}\sin 2\theta =4$
Prove that the following curves intersect orthogonally
$r=a\sin \theta $ and $r=a\cos \theta $
$r=a\left( 1-\cos \theta \right)$ and $r=a\left( 1+\cos \theta \right)$
$r=a\left( 1-\sin \theta \right)$ and $r=a\left( 1+\sin \theta \right)$
$\frac{a}{r}=1+\cos \theta $ and $\frac{b}{r}=1-\cos \theta $
${{r}^{m}}={{a}^{m}}\cos m\theta \text{ }$ and ${{r}^{m}}={{b}^{m}}\sin m\theta \text{ }$
Find the length of polar subtangent for the following curves
$r=a(1-\cos \theta )\text{ }$
$r=a(1+\cos \theta )\text{ }$
Find the length of polar subnormal for the following curves
$r=a(1+\cos \theta )\text{ }$
${{r}^{2}}={{a}^{2}}\cos 2\theta \text{ }$
Show that for the curve $r=a\theta $, polar subnormal is constant.
Show that for the curve $r\theta =a$, polar subtangent is constant.
Show that the length of the polar tangent is constant for the cure
$\theta ={{\cos }^{-1}}\left( {r}/{a}\; \right)-\left( {1}/{r}\; \right)\sqrt{a{}^{2}-{{r}^{2}}}$
Show that the locus of the extremity of the polar subtangent of the curve $\frac{1}{r}=f(\theta )$ is $\frac{1}{r}+f\left( \theta +\frac{\pi }{2} \right)=0$
Find the length of the perpendicular from the pole on a tangent to the curve $r(\theta -1)=a{{\theta }^{2}}$
Find the pedal equation of the curve $r=a(1+\cos \theta )\text{ }$
Find the pedal equation of the curve ${{r}^{2}}={{a}^{2}}\cos 2\theta \text{ }$
Find the pedal equation of the curve ${{r}^{2}}\cos 2\theta \text{ }={{a}^{2}}$
Find the pedal equation of the curve $r=a\operatorname{sech}(n\theta )\text{ }$
Find the pedal equation of the curve ${{r}^{m}}={{a}^{m}}\cos m\theta \text{ }$
Find the pedal equation of the curve ${{r}^{n}}={{a}^{n}}\sin n\theta \text{ }$
Find the pedal equation of the curve $r\theta =a\text{ }$
Find the pedal equation of the curve $r=a\theta \text{ }$
Find the pedal equation of the curve ${{x}^{\text{2}}}+{{y}^{2}}=2ax$
Find the pedal equation of the curve ${{x}^{\text{2}}}-{{y}^{2}}={{a}^{2}}$
Find the pedal equation of the curve $\frac{{{x}^{\text{2}}}}{{{a}^{2}}}-\frac{{{y}^{2}}}{{{b}^{2}}}=1$
For the parabola ${{y}^{2}}=4ax$, prove that \[\frac{ds}{dx}=\sqrt{\frac{a+x}{x}}\]
For the curve $x=a(1-\cos t)\text{ }$$y=a(t+\sin t)\text{ }$, find \[\frac{ds}{dt}\],\[\frac{ds}{dx}\]\[\frac{ds}{dy}\]
Find \[\frac{ds}{d\theta }\] for the curve $r=a(1+\cos \theta )\text{ }$
Find \[\frac{ds}{d\theta }\] for the curve $\frac{2a}{r}=1+\cos \theta $
Find \[\frac{ds}{dt}\] for the curve \[x\sin t+y\cos t=f'(t)\] , \[x\cos t-y\sin t=f''(t)\]
Prove \[\frac{ds}{d\theta }=\frac{{{r}^{2}}}{p}\text{ and }\frac{ds}{dr}=\frac{r}{\sqrt{{{r}^{2}}-{{p}^{2}}}}\]
For the curve ${{r}^{m}}={{a}^{m}}\cos m\theta \text{ }$, prove that
\[\]\[\frac{ds}{d\theta }=\frac{{{a}^{m}}}{{{r}^{m-1}}}\text{=}a{{\left( \sec m\theta \right)}^{{}^{m-1}/{}_{m}}}\text{ }\]
\[{{a}^{2m}}\frac{{{d}^{2}}r}{d{{s}^{2}}}+m{{r}^{2m-1}}=0\]
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.
If $r=a{{e}^{\theta \cot \alpha }}$ then prove that \[(a)\text{ }s=cr\text{ }(b)\text{ }\frac{dr}{ds}=\cos \alpha \]
For any curve prove that \[{{\sin }^{2}}\phi \frac{d\phi }{d\theta }+r\frac{{{d}^{2}}r}{d{{s}^{2}}}=0\]
If $\frac{2a}{r}=1+\cos \theta $ then with usual notation show that \[\frac{ds}{d\psi }=\frac{2a}{{{\sin }^{3}}\psi }\]
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