Question

Solve the equations

$\sf \: \sqrt{ {x}^{2} + 12y } + \sqrt{ {y}^{2} + 12x} = 33$

and

$\sf \: x + y = 23$


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Answer 1

$\large\underline{\sf{Solution-}}$

Given that,

$\red{\rm :\longmapsto\:\sf \: x + y = 23}$

and

$\rm :\longmapsto\: \: \sqrt{ {x}^{2} + 12y } + \sqrt{ {y}^{2} + 12x} = 33$

can be rewritten as

$\rm :\longmapsto\: \: \sqrt{ {x}^{2} + 12(23 - x) } + \sqrt{ {(23 - x)}^{2} + 12x} = 33$

$\red{\bigg \{ \because \: y \: = \: 23 \: - \: x\bigg \}}$

$\rm :\longmapsto\: \sqrt{ {x}^{2} - 12x + 276} + \sqrt{529 + {x}^{2} - 46x + 12x} = 33$

$\rm :\longmapsto\: \sqrt{ {x}^{2} - 12x + 276} + \sqrt{529 + {x}^{2} - 34x} = 33$

Let assume that,

$\rm :\longmapsto\:a = \sqrt{ {x}^{2} - 12x + 276} - - - - (1)$

and

$\rm :\longmapsto\:b = \sqrt{ {x}^{2} - 34x + 529} - - - - - (2)$

So, that,

$\red{\rm :\longmapsto\:a + b = 33} - - - - (3)$

and

$\red{\rm :\longmapsto\: {a}^{2} - {b}^{2}}$

$\rm \:  =  \: {x}^{2} - 12x + 276 - {x}^{2} + 34x - 529$

$\rm \:  =  \: 22x - 253$

$\rm \:  =  \: 11(2x - 23)$

$\red{\bf\implies \: {a}^{2} - {b}^{2} = 11(2x - 23)}$

$\rm :\longmapsto\:(a + b)(a - b) = 11(2x - 23)$

$\rm :\longmapsto\:33(a - b) = 11(2x - 23)$

$\rm :\longmapsto\:a - b = \dfrac{2x - 23}{3} - - - - (4)$

On adding equation (3) and (4), we get

$\rm :\longmapsto\:2a = 33 + \dfrac{2x - 23}{3}$

$\rm :\longmapsto\:2a = \dfrac{99 + 2x - 23}{3}$

$\rm :\longmapsto\:2a = \dfrac{76 + 2x}{3}$

$\rm :\longmapsto\:a = \dfrac{38 + x}{3} - - - - (5)$

On substituting the value of a in equation (1), we get

$\rm :\longmapsto\:\dfrac{38 + x}{3} = \sqrt{ {x}^{2} - 12x + 276}$

$\rm :\longmapsto\:38 + x = 3\sqrt{ {x}^{2} - 12x + 276}$

On squaring both sides, we get

$\rm :\longmapsto\: {(38 + x)}^{2} = 9( {x}^{2} - 12x + 276)$

$\rm :\longmapsto\:1444 + {x}^{2} + 76x = 9 {x}^{2} - 108x + 2128$

$\rm :\longmapsto\: {8x}^{2} - 184x + 1040 = 0$

$\rm :\longmapsto\: 8({x}^{2} - 23x + 130) = 0$

$\rm :\longmapsto\: {x}^{2} - 23x + 130 = 0$

$\rm :\longmapsto\: {x}^{2} - 13x - 10x + 130 = 0$

$\rm :\longmapsto\:x(x - 13) - 10(x - 13) = 0$

$\rm :\longmapsto\:(x - 13)(x - 10) = 0$

$\bf\implies \:x = 13 \: \: \: \: or \: \: \: \: x = 10$

So, corresponding values of y are

$\purple{\begin{gathered}\boxed{\begin{array}{c|c} \bf x & \bf y \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf 10 & \sf 13 \\ \\ \sf 13 & \sf 10 \end{array}} \\ \end{gathered}}$