The-line-y-mx-c-touches-ellipse-x-2-a-2-y-2-b-2-1-prove-that-the-foot-of-perpendicular-from-focus-into-this-line-lie-on-auxillary-circle-x-2-y-2-a-2-

Question Number 51590 by peter frank last updated on 28/Dec/18

T h e l i n e y = m x + c t o u c h e s

e l l i p s e \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1

p r o v e t h a t t h e f o o t o f

p e r p e n d i c u l a r f r o m

f o c u s i n t o t h i s l i n e l i e o n

a u x i l l a r y c i r c l e

x^{2} + y^{2} = a^{2}

Commented by Necxx last updated on 28/Dec/18

m r p e t e r f r a n k w h i c h c o u n t r y d o

y o u h a i l f r o m ? ?

Answered by tanmay.chaudhury50@gmail.com last updated on 28/Dec/18

c^{2} = a^{2} m^{2} + b^{2} \leftarrow

f o c u s (c_{1}, 0) a n d (- c_{1}, 0)

c_{1}^{2} = a^{2} - b^{2} \leftarrow

t h e e q n o f s t l i n e p a s s i n g t h r o u g h (c_{1}, 0) a n d

⊥ t o y = m x + c i s

y - 0 = \frac{- 1}{m} (x - c_{1})

y = \frac{- x}{m} + \frac{c_{1}}{m}

s o l v e y = m x + c a n d y = \frac{- x}{m} + \frac{c_{1}}{m} t o f i n d f o o t o f

p e r p e n d i c u l a r

y = m x + c

y = \frac{- x}{m} + \frac{c_{1}}{m}

m x + c = \frac{- x}{m} + \frac{c_{1}}{m}

x (m + \frac{1}{m}) = \frac{c_{1}}{m} - c

x = \frac{c_{1} - m c}{m^{2} + 1} s o y = m (\frac{c_{1} - m c}{m^{2} + 1}) + c

x = \frac{c_{1} - m c}{m^{2} + 1} y = \frac{{m c}_{1} - m^{2} c + m^{2} c + c}{m^{2} + 1}

x^{2} + y^{2} = \frac{c_{1}^{2} - 2 {m c c}_{1} + m^{2} c^{2} + m^{2} c_{1}^{2} + 2 {m c c}_{1} + c^{2}}{{(m^{2} + 1)}^{2}}

= \frac{c_{1}^{2} (m^{2} + 1) + c^{2} (m^{2} + 1)}{{(m^{2} + 1)}^{2}}

= \frac{(m^{2} + 1)}{{(m^{2} + 1)}^{2}} \times (c^{2} + c_{1}^{2})

= \frac{a^{2} m^{2} + b^{2} + a^{2} - b^{2}}{(m^{2} + 1)}

= a^{2}

Commented by peter frank last updated on 28/Dec/18

t h a n k y o u s i r

Answered by peter frank last updated on 28/Dec/18

y = m x + c

P = (x, y)

S = (a e, 0)

e q u a t i o n = ?

s l o p e o f n o r m a l M

s l o p e o f t a n g e n t - \frac{1}{m}

- \frac{1}{m} = \frac{y}{x - a e}

a e = m y + x

a^{2} e^{2} = m^{2} y^{2} + 2 m x y + x^{2} \dots (i)

r e c a l l

y = m x + \sqrt{a^{2} m^{2} + b^{2}}

y - m x = \sqrt{a^{2} m^{2} + b^{2}}

y^{2} - 2 m x y + m^{2} x^{2} = a^{2} m^{2} + b^{2} \dots . (i i)

a d d (i) + (i i)

a^{2} e^{2} = m^{2} y^{2} + 2 m x y + x^{2} =

y^{2} - 2 m x y + m^{2} x^{2} = a^{2} m^{2} + b^{2}

x^{2} + y^{2} = a^{2}