If-lx-my-1-touches-the-curve-ax-n-by-n-1-show-that-l-a-n-n-1-m-b-n-n-1-1-

Question Number 84067 by niroj last updated on 09/Mar/20

If l x + m y = 1 touches the curve {(ax)}^{n} + {(by)}^{n} = 1, show that

{(\frac{l}{a})}^{\frac{n}{n - 1}} + {(\frac{m}{b})}^{\frac{n}{n - 1}} = 1 .

Answered by mind is power last updated on 09/Mar/20

l = m = a = b \Rightarrow

1 + 1 = 2

x + y = 1

x^{2} + y^{2} = 1 \Rightarrow x y = 0

t h e Q u a t i o n i s T a n g e n t e ?

Answered by som(math1967) last updated on 09/Mar/20

l e t a t (p_{,} q) l i n e t o u c h e s t h e c u r v e

∴ {(a p)}^{n} + {(b q)}^{n} = 1

s l o p e o f l i n e

{(\frac{d y}{d x})}_{p, q} = \frac{- a^{n} p^{n - 1}}{b^{n} q^{n - 1}}

e q u n . o f l i n e

y - q = \frac{- a^{n} p^{n - 1}}{b^{n} q^{n - 1}} (x - p)

a^{n} p^{n - 1} x + b^{n} q^{n - 1} y = {(a p)}^{n} + {(b q)}^{n} = 1

[{(a p)}^{n} + {(b q)}^{n} = 1]

N o w l x + m y = 1 a n d

a^{n} p^{n - 1} x + b^{n} q^{n - 1} y = 1 a r e s a m e

s t . l i n e ∴ \frac{a^{n} p^{n - 1}}{l} = \frac{b^{n} q^{n - 1}}{m} = \frac{1}{1}

∴ p = {(\frac{l}{a^{n}})}^{\frac{1}{n - 1}} q = {(\frac{m}{b^{n}})}^{\frac{1}{n - 1}}

n o w l p + m q = 1

[l i n e t o u c h e s a t (p, q)]

\frac{l \times l^{\frac{1}{n - 1}}}{a^{\frac{n}{n - 1}}} + \frac{m \times m^{\frac{1}{n - 1}}}{b^{\frac{n}{n - 1}}} = 1

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

Commented by niroj last updated on 09/Mar/20

g o o d w o r k .

Answered by TANMAY PANACEA last updated on 09/Mar/20

m y = 1 - l x

y = (\frac{- l}{m}) x + \frac{1}{m}

s l o p e = (\frac{- l}{m}) a n d p o i n t o f t a n g e n t (α, β)

l α + m β = 1 \dots

{(a x)}^{n} + {(b y)}^{n} = 1

a^{n} \times {n x}^{n - 1} + b^{n} \times {n y}^{n - 1} \times \frac{d y}{d x} = 0

\frac{d y}{d x} = \frac{- {a x}^{n - 1}}{{b y}^{n - 1}} = (- 1) (\frac{a}{b}) {(\frac{x}{y})}^{n - 1}

(- 1) (\frac{a}{b}) {(\frac{α}{β})}^{n - 1} = (\frac{- l}{m})

{(\frac{α}{β})}^{n - 1} = (\frac{l b}{m a})

\frac{α}{β} = {(\frac{l b}{m a})}^{\frac{1}{n - 1}}

\frac{α}{{(l b)}^{\frac{1}{n - 1}}} = \frac{β}{{(m a)}^{\frac{1}{n - 1}}} = k (s a y)

α = k . {(l b)}^{\frac{1}{n - 1}} a n d β = k . {(m a)}^{\frac{1}{n - 1}}

s o

{(a α)}^{n} + {(b β)}^{n} = 1

{a . k . {(l b)}^{\frac{1}{n - 1}}}^{n} + {b . k . {(m a)}^{\frac{1}{n - 1}}}^{n} = 1

k^{n} \times [a^{n} . {(l b)}^{\frac{n}{n - 1}} + b^{n} . {(m a)}^{\frac{n}{n - 1}}] = 1

l α + m β = 1

l \times k {(l b)}^{\frac{1}{n - 1}} + m \times k {(m a)}^{\frac{1}{n - 1}} = 1

w a i t p l s