Question-127509

Question Number 127509 by mr W last updated on 30/Dec/20

Commented by mr W last updated on 31/Dec/20

i f b o t h e l l i p s e s h a v e t h e s a m e s h a p e

a s \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1, d e t e r m i n e \frac{b}{a} = ?

Commented by mr W last updated on 28/Jan/21

Answered by mr W last updated on 31/Dec/20

Commented by mr W last updated on 25/Jan/21

l e t m = \tan θ, μ = \frac{b}{a}

\Rightarrow \sin θ = \frac{m}{\sqrt{1 + m^{2}}}, \cos θ = \frac{1}{\sqrt{1 + m^{2}}}

i n x^{'} y^{'} - s y s t e m :

O (h, k)

e q n . o f O A :

y = k + m (x - h)

\Rightarrow m x - y + k - m h = 0

O A i s t a n g e n t t o e l l i p s e :

a^{2} m^{2} + b^{2} = {(k - m h)}^{2}

\Rightarrow k - m h = - \sqrt{a^{2} m^{2} + b^{2}} \dots (i)

e q n . o f O B :

y = k - \frac{1}{m} (x - h)

\Rightarrow \frac{x}{m} + y - k - \frac{h}{m} = 0

O B i s t a n g e n t t o e l l i p s e :

\frac{a^{2}}{m^{2}} + b^{2} = {(k + \frac{h}{m})}^{2}

\Rightarrow k + \frac{h}{m} = - \sqrt{\frac{a^{2}}{m^{2}} + b^{2}} \dots (i i)

f r o m (i) a n d (i i) :

\Rightarrow \frac{h}{a} = - \frac{1}{1 + m^{2}} (\sqrt{1 + m^{2} μ^{2}} - m \sqrt{m^{2} + μ^{2}})

\Rightarrow \frac{k}{a} = - \frac{1}{1 + m^{2}} (m \sqrt{1 + m^{2} μ^{2}} + \sqrt{m^{2} + μ^{2}})

s a y P (a \cos ϕ, - b \sin ϕ)

\tan (θ - φ) = \frac{d y}{d x} = \frac{- b \cos ϕ}{- a \sin ϕ} = \frac{μ}{\tan ϕ}

\frac{m - \tan φ}{1 + m \tan φ} = \frac{μ}{\tan ϕ}

l e t η = \tan ϕ

\sin ϕ = \frac{η}{\sqrt{1 + η^{2}}}, \cos ϕ = \frac{1}{\sqrt{1 + η^{2}}}

\Rightarrow \tan φ = \frac{η - \frac{μ}{m}}{\frac{η}{m} + μ}

i n x y - s y s t e m :

s a y P (x_{p}, y_{P})

x_{P} = (- k - b \sin ϕ) \sin θ + (- h + a \cos ϕ) \cos θ

\frac{x_{P}}{a} = [\frac{1}{1 + m^{2}} (m \sqrt{1 + m^{2} μ^{2}} + \sqrt{m^{2} + μ^{2}}) - μ \sin ϕ] \sin θ + [\frac{1}{1 + m^{2}} (\sqrt{1 + m^{2} μ^{2}} - m \sqrt{m^{2} + μ^{2}}) + \cos ϕ] \cos θ

\frac{x_{P}}{a} = \frac{m}{\sqrt{1 + m^{2}}} [\frac{1}{1 + m^{2}} (m \sqrt{1 + m^{2} μ^{2}} + \sqrt{m^{2} + μ^{2}}) - \frac{μ η}{\sqrt{1 + η^{2}}}] + \frac{1}{\sqrt{1 + m^{2}}} [\frac{1}{1 + m^{2}} (\sqrt{1 + m^{2} μ^{2}} - m \sqrt{m^{2} + μ^{2}}) + \frac{1}{\sqrt{1 + η^{2}}}]

\Rightarrow \frac{x_{P}}{a} = \frac{1}{\sqrt{1 + m^{2}}} (\sqrt{1 + m^{2} μ^{2}} + \frac{1 - m μ η}{\sqrt{1 + η^{2}}})

y_{P} = (- k - b \sin ϕ) \cos θ - (- h + a \cos ϕ) \sin θ

\frac{y_{P}}{a} = [\frac{1}{1 + m^{2}} (m \sqrt{1 + m^{2} μ^{2}} + \sqrt{m^{2} + μ^{2}}) - μ \sin ϕ] \cos θ - [\frac{1}{1 + m^{2}} (\sqrt{1 + m^{2} μ^{2}} - m \sqrt{m^{2} + μ^{2}}) + \cos ϕ] \sin θ

\frac{y_{P}}{a} = \frac{1}{\sqrt{1 + m^{2}}} [\frac{1}{1 + m^{2}} (m \sqrt{1 + m^{2} μ^{2}} + \sqrt{m^{2} + μ^{2}}) - \frac{μ η}{\sqrt{1 + η^{2}}}] - \frac{m}{\sqrt{1 + m^{2}}} [\frac{1}{1 + m^{2}} (\sqrt{1 + m^{2} μ^{2}} - m \sqrt{m^{2} + μ^{2}}) + \frac{1}{\sqrt{1 + η^{2}}}]

\Rightarrow \frac{y_{P}}{a} = \frac{1}{\sqrt{1 + m^{2}}} (\sqrt{m^{2} + μ^{2}} - \frac{m + μ η}{\sqrt{1 + η^{2}}})

\frac{x_{P}^{2}}{a^{2}} + \frac{y_{P}^{2}}{b^{2}} = 1

\Rightarrow {(\sqrt{1 + m^{2} μ^{2}} + \frac{1 - m μ η}{\sqrt{1 + η^{2}}})}^{2} + \frac{1}{μ^{2}} {(\sqrt{m^{2} + μ^{2}} - \frac{m + μ η}{\sqrt{1 + η^{2}}})}^{2} = 1 + m^{2} \dots (I)

\tan φ = - \frac{d y}{d x} = {(\frac{b}{a})}^{2} \frac{x_{P}}{y_{P}}

\frac{η - \frac{μ}{m}}{\frac{η}{m} + μ} = μ^{2} \times \frac{\sqrt{1 + m^{2} μ^{2}} + \frac{1 - m μ η}{\sqrt{1 + η^{2}}}}{\sqrt{m^{2} + μ^{2}} - \frac{m + μ η}{\sqrt{1 + η^{2}}}}

\Rightarrow μ^{2} (\frac{η}{m} + μ) (\sqrt{1 + m^{2} μ^{2}} + \frac{1 - m μ η}{\sqrt{1 + η^{2}}}) = (η - \frac{μ}{m}) (\sqrt{m^{2} + μ^{2}} - \frac{m + μ η}{\sqrt{1 + η^{2}}}) \dots (I I)

f o r a n y v a l u e i n r a n g e 0 < θ < 60 ° w e

c a n f i n d t h e c o r e s p o n d i n g μ f r o m

(I) a n d (I I) .

w e g e t μ_{m a x} \approx 0.38436919447 a t θ \approx 37.9382 °

Commented by mr W last updated on 31/Dec/20

Commented by mr W last updated on 31/Dec/20

Commented by mr W last updated on 31/Dec/20

Commented by mr W last updated on 02/Jan/21

Answered by mr W last updated on 30/Jan/21

Commented by mr W last updated on 30/Jan/21

l e t m = \tan θ, μ = \frac{b}{a}

\Rightarrow \sin θ = \frac{m}{\sqrt{1 + m^{2}}}, \cos θ = \frac{1}{\sqrt{1 + m^{2}}}

i n x^{'} y^{'} - s y s t e m :

O (h, k)

e q n . o f O A :

y = k + m (x - h)

\Rightarrow m x - y + k - m h = 0

O A i s t a n g e n t t o e l l i p s e :

a^{2} m^{2} + b^{2} = {(k - m h)}^{2}

\Rightarrow k - m h = - \sqrt{a^{2} m^{2} + b^{2}} \dots (i)

e q n . o f O B :

y = k - \frac{1}{m} (x - h)

\Rightarrow \frac{x}{m} + y - k - \frac{h}{m} = 0

O B i s t a n g e n t t o e l l i p s e :

\frac{a^{2}}{m^{2}} + b^{2} = {(k + \frac{h}{m})}^{2}

\Rightarrow k + \frac{h}{m} = - \sqrt{\frac{a^{2}}{m^{2}} + b^{2}} \dots (i i)

f r o m (i) a n d (i i) :

\Rightarrow \frac{h}{a} = - \frac{1}{1 + m^{2}} (\sqrt{1 + m^{2} μ^{2}} - m \sqrt{m^{2} + μ^{2}})

\Rightarrow \frac{k}{a} = - \frac{1}{1 + m^{2}} (m \sqrt{1 + m^{2} μ^{2}} + \sqrt{m^{2} + μ^{2}})

i n x y - s y s t e m :

s a y O^{'} (x_{C}, y_{C})

x_{C} = - k \sin θ - h \cos θ

\frac{x_{C}}{a} = \frac{1}{1 + m^{2}} (\sqrt{1 + m^{2} μ^{2}} - m \sqrt{m^{2} + μ^{2}}) \cos θ + \frac{1}{1 + m^{2}} (m \sqrt{1 + m^{2} μ^{2}} + \sqrt{m^{2} + μ^{2}}) \sin θ

\Rightarrow \frac{x_{C}}{a} = \frac{\sqrt{1 + m^{2} μ^{2}}}{\sqrt{1 + m^{2}}}

y_{C} = - k \cos θ + h \sin θ

\frac{y_{C}}{a} = - \frac{1}{1 + m^{2}} (\sqrt{1 + m^{2} μ^{2}} - m \sqrt{m^{2} + μ^{2}}) \sin θ + \frac{1}{1 + m^{2}} (m \sqrt{1 + m^{2} μ^{2}} + \sqrt{m^{2} + μ^{2}}) \cos θ

\Rightarrow \frac{y_{C}}{a} = \frac{\sqrt{m^{2} + μ^{2}}}{\sqrt{1 + m^{2}}}

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

\Rightarrow μ^{2} ξ^{2} + η^{2} = μ^{2} \dots (i)

\Rightarrow η = μ \sqrt{1 - ξ^{2}}

e q n . o f e l l i p s e O^{'} :

\frac{{[(x - x_{C}) \frac{1}{\sqrt{1 + m^{2}}} - (y - y_{C}) \frac{m}{\sqrt{1 + m^{2}}} + x_{C}]}^{2}}{a^{2}} + \frac{{[(x - x_{C}) \sin θ - (y - y_{C}) \cos θ + y_{C}]}^{2}}{b^{2}} = 1

{[(\frac{x}{a} - \frac{\sqrt{1 + m^{2} μ^{2}}}{\sqrt{1 + m^{2}}}) - m (\frac{y}{a} - \frac{\sqrt{m^{2} + μ^{2}}}{\sqrt{1 + m^{2}}}) + \sqrt{1 + m^{2} μ^{2}}]}^{2} + \frac{{[m (\frac{x}{a} - \frac{\sqrt{1 + m^{2} μ^{2}}}{\sqrt{1 + m^{2}}}) + (\frac{y}{a} - \frac{\sqrt{m^{2} + μ^{2}}}{\sqrt{1 + m^{2}}}) + \sqrt{m^{2} + μ^{2}}]}^{2}}{μ^{2}} = 1 + m^{2}

\Rightarrow μ^{2} {[ξ - m η + \frac{m \sqrt{m^{2} + μ^{2}} - \sqrt{1 + m^{2} μ^{2}}}{\sqrt{1 + m^{2}}} + \sqrt{1 + m^{2} μ^{2}}]}^{2} + {[m ξ + η - \frac{\sqrt{m^{2} + μ^{2}} + m \sqrt{1 + m^{2} μ^{2}}}{\sqrt{1 + m^{2}}} + \sqrt{m^{2} + μ^{2}}]}^{2} = μ^{2} (1 + m^{2}) \dots (i i)

\Rightarrow μ^{2} {(ξ - m η + p)}^{2} + {(m ξ + η + q)}^{2} = μ^{2} (1 + m^{2}) \dots (i i)

\dots \dots

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