Question-45630

Question Number 45630 by ajfour last updated on 14/Oct/18

Answered by tanmay.chaudhury50@gmail.com last updated on 15/Oct/18

\frac{1}{2} \times l \times x = \sqrt{s (s - a) (s - b) (s - c)}

s = \frac{R + x + x + r + l}{2} = x + \frac{R + r + l}{2}

a = R + x b = x + r c = l

\frac{1}{2} l x = \sqrt{(x + \frac{R + r + l}{2}) (\frac{r + l + R}{2} - R) (\frac{R + l + r}{2} - r) (x + \frac{R + r + l}{2} - l)}

\frac{l^{2} x^{2}}{4} = (x + M) (N) (P) (x + Q) ↑ h e r e i f o r g o t t o p u t x

l^{2} x^{2} = 4 (x + M) N P (x + Q) M = \frac{R + r + l}{2} N = (\frac{R + r + l}{2} - R)

P = (\frac{R + r + l}{2} - r) Q = \frac{R + r + l}{2} - l

\frac{x^{2}}{(x + M) (x + Q)} = \frac{4 N P}{l^{2}}

n w p u t M, N, P, Q v a l u e

\frac{x^{2}}{(x + M) (x + Q)} = T

x^{2} = T (x^{2} + Q x + M x + M Q)

x^{2} (T - 1) + x (Q + M) + M Q T = 0

x = \frac{- (Q + M) + \sqrt{{(Q + M)}^{2} - 4 (T - 1) (M Q T)}}{2 (T - 1)}

Commented by ajfour last updated on 15/Oct/18

t h a n k s, T a n m a y S i r .

Answered by MrW3 last updated on 15/Oct/18

\sqrt{{(x + r)}^{2} - x^{2}} + \sqrt{{(R + x)}^{2} - x^{2}} = l

\sqrt{r (2 x + r)} + \sqrt{R (2 x + R)} = l

r (2 x + r) + R (2 x + R) + 2 \sqrt{r R (2 x + r) (2 x + R)} = l^{2}

2 \sqrt{r R (2 x + r) (2 x + R)} = (l^{2} - R^{2} - r^{2}) - 2 (R + r) x

l e t 2 e^{2} = l^{2} - R^{2} - r^{2} o r e = \sqrt{\frac{l^{2} - R^{2} - r^{2}}{2}}

\Rightarrow \sqrt{r R (2 x + r) (2 x + R)} = e^{2} - (R + r) x

r R (2 x + r) (2 x + R) = e^{4} + {(R + r)}^{2} x^{2} - 2 e^{2} (R + r) x

r R {4 x^{2} + 2 (R + r) x + R r} = e^{4} + {(R + r)}^{2} x^{2} - 2 e^{2} (R + r) x

4 {R r x}^{2} + 2 R r (R + r) x + R^{2} r^{2} = e^{4} + {(R + r)}^{2} x^{2} - 2 e^{2} (R + r) x

{{(R + r)}^{2} - 4 R r} x^{2} - 2 (R + r) (e^{2} + R r) x - (R^{2} r^{2} - e^{4}) = 0

{(R - r)}^{2} x^{2} - 2 (R + r) (e^{2} + R r) x - (R^{2} r^{2} - e^{4}) = 0

f o r R = r :

\Rightarrow x = \frac{e^{4} - R^{4}}{4 R (e^{2} + R^{2})} = \frac{e^{2} - R^{2}}{4 R} = \frac{l^{2} - 4 R^{2}}{8 R}

f o r R \neq r :

\Rightarrow x = \frac{2 (R + r) (e^{2} + R r) - 2 \sqrt{{(R + r)}^{2} {(e^{2} + R r)}^{2} + {(R - r)}^{2} (R^{2} r^{2} - e^{4})}}{{(R - r)}^{2}}

\Rightarrow x = \frac{(R + r) (2 e^{2} + 2 R r) - 2 \sqrt{R r (2 e^{2} + 2 R r) (R^{2} + r^{2} + 2 e^{2})}}{2 {(R - r)}^{2}}

\Rightarrow x = \frac{(R + r) [l^{2} - {(R - r)}^{2}] - 2 l \sqrt{R r [l^{2} - {(R - r)}^{2}]}}{2 {(R - r)}^{2}}

Commented by ajfour last updated on 15/Oct/18

t h a n k y o u S i r!

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