If P equiv left( frac{1}{x_p}, p right), Q = | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) Let the $k^{th}$ term of the $H.P.$ be $x_k$. By definition,$\frac{1}{x_k}$ is the $k^{th}$ term of an $A.P.$Let $a_k = \frac{1}{x_k} = a + (k-1)d

Vedclass · Accepted Answer

the points $P, Q, R$ are collinear

Vedclass · Answer

(C) Let the $k^{th}$ term of the $H.P.$ be $x_k$. By definition,$\frac{1}{x_k}$ is the $k^{th}$ term of an $A.P.$Let $a_k = \frac{1}{x_k} = a + (k-1)d$,where $a$ is the first term and $d$ is the common difference.The coordinates of the points are $P(a_p, p)$,$Q(a_q, q)$,and $R(a_r, r)$.The slope of line segment $PQ$ is $m_{PQ} = \frac{q - p}{a_q - a_p} = \frac{q - p}{(a + (q-1)d) - (a + (p-1)d)} = \frac{q - p}{(q - p)d} = \frac{1}{d}$.The slope of line segment $QR$ is $m_{QR} = \frac{r - q}{a_r - a_q} = \frac{r - q}{(a + (r-1)d) - (a + (q-1)d)} = \frac{r - q}{(r - q)d} = \frac{1}{d}$.Since the slopes $m_{PQ} = m_{QR} = \frac{1}{d}$,the points $P, Q,$ and $R$ lie on the same line.Therefore,the points $P, Q, R$ are collinear.

If $P \equiv \left( \frac{1}{x_p}, p \right), Q = \left( \frac{1}{x_q}, q \right), R = \left( \frac{1}{x_r}, r \right)$ where $x_k \neq 0$ denotes the $k^{th}$ term of an $H.P.$ for $k \in N$,then:

Similar Questions

For $c \neq 0, c \neq 1$,if the straight lines $x+y=1$,$2x-y=c$,and $bx+2by=c$ have one common point,then:

The equation of the line passing through the point of intersection of the lines $4x - 3y - 1 = 0$ and $5x - 2y - 3 = 0$ and parallel to the line $2y - 3x + 2 = 0$ is:

If the three lines $2x + y = 4$,$3x + 2y = 3$,and $ax + 3y = 2$ are concurrent,find the value of $a$.

The value of $\lambda$ for which the lines $3x + 4y = 5,$ $5x + 4y = 4,$ and $\lambda x + 4y = 6$ meet at a point is

What is the equation of the line passing through the intersection of the lines $3x - 4y + 1 = 0$ and $5x + y - 1 = 0$ and having equal intercepts on the axes?

Vedclass Test Series

Exam Paper Generator

Online Exam Module