Question-207018

Question Number 207018 by cherokeesay last updated on 03/May/24

Answered by mr W last updated on 03/May/24

Commented by cherokeesay last updated on 03/May/24

R = ?

Commented by mr W last updated on 03/May/24

4 z^{2} = 2 \times 10^{2} + 2 \times 9^{2} - {(2 R)}^{2}

\Rightarrow z^{2} = \frac{181}{2} - R^{2}

{B F}^{2} = x^{2} + R^{2}

{C F}^{2} = y^{2} + R^{2}

4 z^{2} + 10 (x^{2} + R^{2}) = 14 (R^{2} + 4 \times 10)

\Rightarrow 4 z^{2} + 10 x^{2} = 4 R^{2} + 560

\Rightarrow 5 x^{2} = 4 R^{2} + 99

5 z^{2} + 9 (y^{2} + R^{2}) = 14 (R^{2} + 5 \times 9)

\Rightarrow 5 z^{2} + 9 y^{2} = 5 R^{2} + 630

\Rightarrow 18 y^{2} = 20 R^{2} + 355

\cos A = \frac{14^{2} + 14^{2} - {(x + y)}^{2}}{2 \times 14 \times 14} = \frac{10^{2} + 9^{2} - {(2 R)}^{2}}{2 \times 10 \times 9}

\Rightarrow 45 {(x + y)}^{2} = 392 R^{2} - 98

\Rightarrow 45 (x^{2} + y^{2} + 2 x y) = 392 R^{2} - 98

\Rightarrow 20 x y = 68 R^{2} - 417

\Rightarrow 20^{2} x^{2} y^{2} = {(68 R^{2} - 417)}^{2}

\Rightarrow 20^{2} \times \frac{4 R^{2} + 99}{5} \times \frac{20 R^{2} + 355}{18} = {(68 R^{2} - 417)}^{2}

\Rightarrow 784 R^{4} - 13192 R^{2} + 3249 = 0

\Rightarrow R = \sqrt{\frac{6596 + \sqrt{6596^{2} - 784 \times 3249}}{784}} = \frac{57}{14} \approx 4.071

Answered by mr W last updated on 04/May/24

Commented by mr W last updated on 03/May/24

\frac{4 \cos θ + 5 \cos θ}{2} = R

\Rightarrow \cos θ = \frac{2 R}{9}

{(2 R)}^{2} = 10^{2} + 9^{2} - 2 \times 10 \times 9 \cos 2 θ

4 R^{2} = 181 - 180 (2 \times \frac{4 R^{2}}{81} - 1)

R^{2} = \frac{3249}{196}

\Rightarrow R = \frac{57}{14} ✓

Commented by cherokeesay last updated on 03/May/24

t h a n k y o u m a s t e r!