If the equation ax^2+bx+c=0 and 5x^2+12x+13=0 have a common root , where a ,b and c are the sides of a triangle ABC , then

Solution:   Since , equation $5x^2+12x+13=0$ has imaginary roots

Hence , equation $ax^{2}+bx+c=0$ and

$5x^{2}+12x+13=0$  have both common roots

$\\ \\ \therefore \hspace{2cm}\frac{a}{5}=\frac{b}{12}=\frac{c}{13}=k(say)\\ \\ \Rightarrow \hspace{2cm}a^{2}+b^{2}=25k^{2}+144k^{2}=(13k)^{2}=c^{2}\\ \\ \Rightarrow \hspace{2cm}a^{2}+b^{2}=c^{2}$

$\Rightarrow$           ABC is right angled triangle.

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