If {x_1}, {x_2}, {x_3} as well as {y_1}, {y_2 | vedclass

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

Question

(A) Given that ${x_1}, {x_2}, {x_3}$ and ${y_1}, {y_2}, {y_3}$ are in $G$.$P$. with the same common ratio $r$.Therefore,we can write: ${x_2} = r{x_1},

Vedclass · Accepted Answer

Lie on a straight line

Vedclass · Answer

(A) Given that ${x_1}, {x_2}, {x_3}$ and ${y_1}, {y_2}, {y_3}$ are in $G$.$P$. with the same common ratio $r$.Therefore,we can write: ${x_2} = r{x_1}, {x_3} = {r^2}{x_1}$ and ${y_2} = r{y_1}, {y_3} = {r^2}{y_1}$.To check if the points $({x_1}, {y_1}), ({x_2}, {y_2}), ({x_3}, {y_3})$ are collinear,we calculate the area of the triangle formed by these points:Area $= \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)|$Substituting the values:Area $= \frac{1}{2} |x_1(ry_1 - r^2y_1) + rx_1(r^2y_1 - y_1) + r^2x_1(y_1 - ry_1)|$Area $= \frac{1}{2} |x_1y_1(r - r^2) + x_1y_1(r^3 - r) + x_1y_1(r^2 - r^3)|$Area $= \frac{1}{2} |x_1y_1(r - r^2 + r^3 - r + r^2 - r^3)| = 0$.Since the area of the triangle is $0$,the points are collinear and lie on a straight line.

If ${x_1}, {x_2}, {x_3}$ as well as ${y_1}, {y_2}, {y_3}$ are in $G$.$P$. with the same common ratio,then the points $({x_1}, {y_1}), ({x_2}, {y_2})$ and $({x_3}, {y_3})$:

Similar Questions

Two sequences ${t_n}$ and ${s_n}$ are defined by $t_n = \log \left( \frac{5^{n+1}}{3^{n-1}} \right)$ and $s_n = \left[ \log \left( \frac{5}{3} \right) \right]^n$. Then:

The least positive integer $n$ such that $1 - \frac{2}{3} - \frac{2}{3^2} - \dots - \frac{2}{3^{n-1}} < \frac{1}{100}$ is:

For all positive integral values of $n$,the value of $3 \cdot 1 \cdot 2 + 3 \cdot 2 \cdot 3 + 3 \cdot 3 \cdot 4 + \dots + 3 \cdot n \cdot (n + 1)$ is

If the first term of a $G.P.$ is $5$ and the common ratio is $-5$,then which term is $3125$ (in $^{th}$)?

How many terms of the $A.P.$ $1, 4, 7, \ldots$ are needed to give the sum $715$?

Vedclass Test Series

Exam Paper Generator

Online Exam Module