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Algebra Difficulty 4.8 AIME Find the answer

18. If the three sides of a right-angled triangle are a,b,c,∠B=90∘a, b, c, \angle B=90^{\circ}, then, the nature of the roots of the equation
a(x2−1)−2cx+b(x2+1)=0 a\left(x^{2}-1\right)-2 c x+b\left(x^{2}+1\right)=0

with respect to xx is:

Pick one

Solution

18. A.
Δ=(2c)2−4(a+b)(b−a)=4(c2+a2−b2)=0. \begin{array}{l} \Delta=(2 c)^{2}-4(a+b)(b-a) \\ =4\left(c^{2}+a^{2}-b^{2}\right)=0 . \end{array}

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.