Question-164728

Question Number 164728 by mr W last updated on 21/Jan/22

Commented by mr W last updated on 21/Jan/22

t w o p o i n t s P a n d Q o n t h e p a r a b o l a

y = \frac{x^{2}}{a} w i t h ∣ P Q ∣= l . t h e n o r m a l l i n e s

a t P a n d Q i n t e r s e c t a t p o i n t S .

f i n d t h e l o c u s o f S .

Answered by ajfour last updated on 21/Jan/22

S (- h, k)

N o r m a l f r o m S

y = - (\frac{a}{2 p}) x + c P (p, \frac{p^{2}}{a}) l i e s o n i t

\frac{p^{2}}{a} = - \frac{a}{2} + c

{(p_{2} - p_{1})}^{2} + \frac{{(p_{2}^{2} - p_{1}^{2})}^{2}}{a^{3}} = L^{2} \dots (I)

k = \frac{a h}{2 p} + \frac{p^{2}}{a} + \frac{a}{2}

\Rightarrow 2 p^{3} + a (a - 2 k) p + a h = 0

w e g e t p_{1}, p_{2}, p_{3} i n t e r m s o f

h, k, a .

s u b s t i t u t i n g i n . . (I) g i v e s r e q u i r e d

t h e t h r e e l o c i i, f o r t h r e e c h o i c e s o f

p a i r s f r o m a m o n g p_{1}, p_{2}, p_{3} .

Commented by mr W last updated on 21/Jan/22

t h a n k s f o r t r y i n g s i r! i^{'} l l a l s o g i v e a

t r y .

Answered by mr W last updated on 21/Jan/22

Commented by mr W last updated on 23/Jan/22

l o c u s i n s h a p e o f a G e r m a n B r e z e l \dots

Commented by mr W last updated on 22/Jan/22

Commented by mr W last updated on 21/Jan/22

s a y P (p, \frac{p^{2}}{a}), Q (q, \frac{q^{2}}{a})

\tan ϕ = \frac{2 p}{a}

\tan φ = \frac{2 q}{a}

{(q - p)}^{2} + {(\frac{q^{2}}{a} - \frac{p^{2}}{a})}^{2} = l^{2}

{(q - p)}^{2} [1 + \frac{{(p + q)}^{2}}{a^{2}}] = l^{2}

[{(q + p)}^{2} - 4 p q] [1 + \frac{{(p + q)}^{2}}{a^{2}}] = l^{2}

l e t u = p + q, v = p q

(u^{2} - 4 v) (1 + \frac{u^{2}}{a^{2}}) = l^{2}

\Rightarrow v = \frac{1}{4} (u^{2} - \frac{a^{2} l^{2}}{a^{2} + u^{2}})

e q n . o f P S :

y = \frac{p^{2}}{a} - \frac{a}{2 p} (x - p)

y = \frac{p^{2}}{a} + \frac{a}{2} - \frac{a x}{2 p}

e q n . o f Q S :

y = \frac{q^{2}}{a} + \frac{a}{2} - \frac{a x}{2 q}

s a y i n t e r s e c t i o n p o i n t S (x_{s}, y_{s})

\frac{p^{2}}{a} + \frac{a}{2} - \frac{{a x}_{s}}{2 p} = \frac{q^{2}}{a} + \frac{a}{2} - \frac{{a x}_{s}}{2 q}

\frac{p^{2}}{a} - \frac{{a x}_{s}}{2 p} = \frac{q^{2}}{a} - \frac{{a x}_{s}}{2 q}

x_{s} = - \frac{2 (q + p) p q}{a^{2}}

\Rightarrow x_{s} = - \frac{2 u v}{a^{2}}

2 y_{s} = \frac{p^{2} + q^{2}}{a} + a - {a x}_{s} (\frac{1}{2 p} + \frac{1}{2 q})

y_{s} = \frac{a}{2} + \frac{p^{2} + q^{2}}{2 a} - \frac{a (p + q) x_{s}}{4 p q}

y_{s} = \frac{a}{2} + \frac{{(p + q)}^{2} - p q}{a}

\Rightarrow y_{s} = \frac{a}{2} + \frac{u^{2} - v}{a}

t a k i n g u a s p a r a m e t e r w e g e t t h e e q n .

o f l o c u s o f p o i n t S :

{\begin{cases} x = - \frac{2 u v}{a^{2}} \\ y = \frac{a}{2} + \frac{u^{2} - v}{a} \end{cases} w i t h v = \frac{1}{4} (u^{2} - \frac{a^{2} l^{2}}{a^{2} + u^{2}})

w i t h u g i v e n, p, q a r e r o o t s o f

z^{2} - u z + \frac{1}{4} (u^{2} - \frac{a^{2} l^{2}}{a^{2} + u^{2}}) = 0

\Rightarrow p, q = \frac{1}{2} (u \mp \frac{a l}{\sqrt{a^{2} + u^{2}}})

Commented by mr W last updated on 21/Jan/22

Commented by mr W last updated on 21/Jan/22

Commented by ajfour last updated on 21/Jan/22

g r e a t w o r k i n g s s i r, i w i l l t r y n

f o l l o w . .

Commented by Tawa11 last updated on 21/Jan/22

Weldone sir

Answered by ajfour last updated on 23/Jan/22

S (h, k)

\frac{q - h}{k - \frac{q^{2}}{a}} = \frac{2 p}{a} \Rightarrow a k - q^{2} = \frac{a^{2}}{2} (\frac{q - h}{p})

\frac{p - h}{k - \frac{p^{2}}{a}} = \frac{2 q}{a} \Rightarrow a k - p^{2} = \frac{a^{2}}{2} (\frac{p - h}{q})

s u b t r a c t i n g

\Rightarrow q^{2} - p^{2} = \frac{a^{2}}{2} (\frac{p}{q} - \frac{q}{p}) + \frac{a^{2} h}{2} (\frac{1}{p} - \frac{1}{q})

\Rightarrow q + p = \frac{a^{2} h}{2 p q} - \frac{a^{2} (q + p)}{2 p q}

\Rightarrow q + p = \frac{a^{2} h}{a^{2} + 2 p q}

s a y q + p = s, p q = m

\Rightarrow s = \frac{a^{2} h}{a^{2} + 2 m} \dots (i)

A d d i n g

2 a k - (p^{2} + q^{2}) = \frac{a^{2}}{2} (\frac{p}{q} + \frac{q}{p}) - \frac{a^{2} h}{2} (\frac{1}{p} + \frac{1}{q})

\Rightarrow 2 a k - s^{2} + 2 m = \frac{a^{2}}{2} (\frac{s^{2} - 2 m}{m} - \frac{h s}{m})

\dots . . (i i)

A n d s^{2} + \frac{s^{2} (s^{2} - 4 m)}{a^{2}} = L^{2} \dots (i i i)

\Rightarrow u s i n g (i) a n d (i i i)

1 + \frac{s^{2} - 4 m}{a^{2}} = \frac{L^{2}}{h^{2}} (1 + \frac{m}{a^{2}})

\Rightarrow m (\frac{L^{2}}{a^{2} h^{2}} + \frac{4}{a^{2}}) = 1 - \frac{L^{2}}{h^{2}}

\Rightarrow m = \frac{a^{2} (1 - \frac{L^{2}}{h^{2}})}{(4 + \frac{L^{2}}{h^{2}})}

\frac{1}{s} = \frac{1}{h} + \frac{2 m}{a^{2} h} = \frac{1}{h} + \frac{2}{h} (\frac{1 - \frac{L^{2}}{h^{2}}}{4 + \frac{L^{2}}{h^{2}}})

N o w s u b s t i t u t i n g f o r m, s i n (i i)

\frac{2 a k}{s^{2}} + 1 + \frac{2 m}{s^{2}} = \frac{a^{2}}{2} (\frac{1}{m} - \frac{2}{s^{2}} - \frac{h}{m s})

{2 a k + 2 a^{2} (\frac{h^{2} - L^{2}}{4 h^{2} + L^{2}}) + a^{2}} {\frac{1}{h} + \frac{2}{h} (\frac{h^{2} - L^{2}}{4 h^{2} + L^{2}})}^{2}

+ 2 = 0

A n d t o m a k e i t c o m p a c t

s a y \frac{h}{L} = X, \frac{k}{L} = Y

\frac{2 a}{L} (Y + \frac{X^{2} - 1}{4 X^{2} + 1} + \frac{a}{L}) {1 + 2 (\frac{X^{2} - 1}{4 X^{2} + 1})}^{2}

+ 2 X^{2} = 0

o r s i m p l y

\frac{X}{(\frac{1}{2} + \frac{X^{2} - 1}{4 X^{2} + 1})} = \pm {(\frac{2 a}{L})}^{1 / 2} \sqrt{\frac{1 - X^{2}}{4 X^{2} + 1} - Y - \frac{a}{L}}

s a y f o r a = 1, L = 2

\frac{x}{{1 + \frac{x^{2} - 4}{2 (x^{2} + 1)}}} = \pm \sqrt{\frac{4 - x^{2}}{4 (x^{2} + 1)} - \frac{y}{2} - \frac{1}{2}}

\Rightarrow y = \frac{4 - x^{2}}{2 (x^{2} + 1)} - \frac{16 {(x^{2} + 1)}^{2}}{{(3 x^{2} - 2)}^{2}} - 1

Commented by ajfour last updated on 23/Jan/22

s o m e l i t t l e e r r o r \dots i s h a l l e d i t

t o d a y . .

Facebook Tweet Pin