QJEEAdv20Ma98e8: JEE Advanced 2020 Mathematics - Class 11 JEEAdv20Ma98e8 Solution: In a triangle PQR, let a = QR, b = RP, and c = PQ. If

It is given that, in a $\Delta$ PQR

PQ = c, QR = a and RP = b, so a + b + c = 0

$\therefore$ |a|² + a . b + a . c = 0 ....(i)

and a . b + |b|² + c . b = 0 ....(ii)

and, also given that |a| = 3, |b| = 4

and ${{a\,.\,c - a\,.\,b} \over {c\,.\,a - c\,.\,b}} = {{|a|} \over {|a| + |b|}} = {3 \over 7}$

$\Rightarrow 7[(a\,.\,c) - (a\,.\,b)] = 3[(c\,.\,a) - (c\,.\,b)]$

from Eqs. (i) and (ii), on putting the values of a . c and c . b, we get,

$\Rightarrow 7[ - 9 - (a\,.\,b) - (a\,.\,b)] = 3[ - 9 - (a\,.\,b) + 16 + (a\,.\,b)]$

$\Rightarrow 7[ - 9 - 2(a\,.\,b)] = 3 \times 7$

$\Rightarrow a\,.\,b = - 6$

$\because$ $|a \times b{|^2} = {(|a||b|)^2} - {(a\,.\,b)^2}$

= $144 - 36 = 108$