Question

If a, b, c are natural numbers such that a² + b² + c² = 29 and ab + bc + ca = 26, and a + b + c = ______.
(a) 9
(b) 6
(c) 7
(d) 10

Answer 1

$$\begin{gathered} \mathrm{Given}: \\ {a}^2+b^2+c^2=29...\left(1\right) \\ ab+bc+ca=26...\left(2\right) \\ \mathrm{Now}, \\ {\left(a+b+c\right)}^2 \\ =a^2+b^2+c^2+2\left(ab+bc+ca\right)\left(\mathrm{Using}\mathrm{the}\mathrm{identity}:{\left(a+b+c\right)}^2=a^2+b^2+c^2+2\left(ab+bc+ca\right)\right) \\ =29+2\left(26\right)\left(\mathrm{From}\left(1\right)\mathrm{and}\left(2\right)\right) \\ =29+52 \\ =81 \\ \mathrm{Since},{\left(a+b+c\right)}^2=81\mathrm{and}a,b,c\mathrm{are}\mathrm{natural}\mathrm{numbers} \\ \mathrm{Therefore},a+b+c=9. \\ \mathrm{Hence},\mathrm{the}\mathrm{correct}\mathrm{option}\mathrm{is}\left(\mathrm{a}\right). \end{gathered}$$