Question 7
If $$a = \sqrt{23a + b}$$, $$b = \sqrt{23b + a}$$, $$a \neq b$$, then the value of $$\sqrt{a^2 + b^2 + 48}$$ is
Solution
Squaring gives $$a^2 = 23a + b$$ and $$b^2 = 23b + a$$. Subtracting, $$(a-b)(a+b) = 22(a-b)$$, and as $$a \neq b$$ this gives $$a + b = 22$$. Adding the two equations, $$a^2 + b^2 = 24(a + b) = 528$$, so $$\sqrt{a^2 + b^2 + 48} = \sqrt{576} = 24$$.
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