Question

If $P\equiv (\frac{1}{x_p},p);Q\equiv (\frac{1}{x_q},q);R\equiv (\frac{1}{x_r},r),$ where $x_k(\ne 0)$ denotes the $k^{th}$ 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 angled triangle
• C. the points P, Q, R are collinear
• D. none of these

Answer 1

Answer: (C)

the points P, Q, R are collinear

If $\frac{1}{x_1},\frac{1}{x_2},\frac{1}{x_3},....\frac{1}{x_k}$ are in HP

then $a_1,a_2,a_3,....a_k$ are in AP

where $\frac{1}{x_1}=a_1,\frac{1}{x_2}=a_2,\frac{1}{x_k}=a_k$

$\frac{1}{x_p}=a_p=a_1+(p-1)d\frac{1}{x_q}=a_q=a_1+(q-1)d\frac{1}{x_r}=a_r=a_1+(r-1)d$

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.