Question

If $P\equiv (\frac{1}{x_p},p);Q=(\frac{1}{x_q},q);R=(\frac{1}{x_r},r)$ where $x_k\ne 0$ denotes the kth term of an HP for $kϵN$ then

• A. $Ar\left(\Delta PQR\right)=\frac{p^2{q}^2{r}^2}{2}\sqrt{{(p-q)}^2+{(q-r)}^2+{(r-p)}^2}$
• B. $\Delta PQR$ is a right angles triangle
• C. the points P, Q, R are collinear
• D. none of these

Answer 1

The correct option is C the points P, Q, R are collinear

Given S = { 1 ,2 ,3 , 5 , 8 , 13 , 21 , 34 } , $\sum max\left(A\right)\Rightarrow$

If $P\equiv (\frac{1}{x_p},P)$ , $Q\equiv (\frac{1}{x_q},q)$ , $R\equiv (\frac{1}{x_r},r)$ where $x_k\ne 0$

We know,

If $\frac{1}{x_1},\frac{1}{x_2},\frac{1}{x_3},---------\frac{1}{x_k}$ are in H.P.

then $a_1,a_2,a_3,---------a_k$ are in A.P.

where $\frac{1}{x_1}=a_1,\frac{1}{x_2}=a_2,$ and $\frac{1}{x_p}=a_p\equiv a_1+(p-1)d.$

$\frac{1}{x_q}=a_q=a_1+(q-1)d,\frac{1}{x_r}=a_r=a_1+(r-1)d.$

slope of $PQ=\frac{(q-p)}{(q-p)d}=\frac{1}{d}$

slope of $QR=\frac{(r-q)}{(r-q)d}=\frac{1}{d}$

slope of $PR=\frac{(r-p)}{(r-p)d}=\frac{1}{d}$

All have same slope, hence all points are collinear.