question_answer

The vertices of the \[\Delta PQR\] are P (0, b), Q (0, 0) and R (a, 0). If the medians PM and QN of PQR are perpendicular, then

A)  \[{{b}^{2}}=2{{a}^{2}}\]               

B)  \[b={{a}^{2}}\]

C)  \[{{a}^{2}}=2{{b}^{2}}\]         

D)         \[a=b\]

E)  \[a=-b\]

Correct Answer: C

Solution :

Given, points are\[P(0,b),Q(0,0)\]and\[R(a,0)\]. Then, coordinates of                 \[M=\left( \frac{0+a}{2},\frac{0+0}{2} \right)=\left( \frac{a}{2},0 \right)\] And        \[N=\left( \frac{0+a}{2},\frac{b+0}{2} \right)=\left( \frac{a}{2},\frac{b}{2} \right)\] Slope of\[PM=\frac{0-b}{a/2-0}\]or\[\frac{-2b}{a}={{m}_{1}}\](say) Slope of\[QN=\frac{b/2-0}{a/2-0}\] \[\Rightarrow \]               \[\frac{b}{a}={{m}_{2}}\]                                              (say) Given,   \[PM\bot QN\] \[\Rightarrow \]               \[{{m}_{1}}.{{m}_{2}}=-1\] \[\Rightarrow \]               \[\frac{+2b}{a}.\frac{b}{a}=-1\] \[\Rightarrow \]               \[{{a}^{2}}=2{{b}^{2}}\]