Question-68783

Question Number 68783 by peter frank last updated on 15/Sep/19

Commented by peter frank last updated on 22/Sep/19

t h a n k y o u

Commented by ajfour last updated on 15/Sep/19

Commented by peter frank last updated on 21/Sep/19

p l e a s e s i r . c o n t i n u e

Commented by ajfour last updated on 21/Sep/19

l e t S^{'} P = a + s, S P = a - s

l e t P (h, k)

\frac{S^{'} P}{S P} = \frac{a + s}{a - s}

a - s = e (\frac{a}{e} - h)

a + s = e (\frac{a}{e} + h)

\Rightarrow s = e h

\frac{S^{'} P}{S P} = \frac{a + e h}{a - e h} \dots \dots \dots \dots \dots . (i)

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \Rightarrow - \frac{d x}{d y} = \frac{a^{2}}{b^{2}} (\frac{y}{x})

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

= - \frac{d x}{d y} ∣_{P} = \frac{a^{2}}{b^{2}} (\frac{k}{h})

e q . o f n o r m a l

y - k = \frac{a^{2}}{b^{2}} (\frac{k}{h}) (x - h)

l e t n o r m a l i n t e r s e c t x - a x i s

a t N (0, x_{i n t})

x_{i n t} = h (1 - \frac{b^{2}}{a^{2}})

\frac{S N}{S^{'} N} = \frac{a e - x_{i n t}}{x_{i n t} + a e} = \frac{a e - h (1 - \frac{b^{2}}{a^{2}})}{h (1 - \frac{b^{2}}{a^{2}}) + a e} = r_{2}

\Rightarrow \frac{S N}{S^{'} N} = \frac{a - e h}{e h + a} \dots \dots \dots \dots . (i i)

f r o m (i) a n d (i i) w e f i n d

\frac{S^{'} P}{S P} = \frac{S^{'} N}{S N}

\Rightarrow P N i s a n g u l a r b i s e c t o r

o f ∠ S^{'} P S .

Commented by peter frank last updated on 22/Sep/19

p l e a s e h e l p 69230

Answered by mr W last updated on 15/Sep/19

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