In a triangle ABC,let G denote its centroid a | vedclass

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

Question

(C) Let $AM = x AB$ and $AN = y AC$. Since $G$ is the centroid,the position vector of $G$ is $\frac{A+B+C}{3}$.Since $M, G, N$ are collinear,there exi

Vedclass · Accepted Answer

$4/9 \leq r < 1/2$

Vedclass · Answer

(C) Let $AM = x AB$ and $AN = y AC$. Since $G$ is the centroid,the position vector of $G$ is $\frac{A+B+C}{3}$.Since $M, G, N$ are collinear,there exists a scalar $k$ such that $\vec{G} = (1-k)\vec{M} + k\vec{N}$.Using the property of the centroid and the given conditions,it can be shown that $\frac{1}{3x} + \frac{1}{3y} = 1$,or $\frac{1}{x} + \frac{1}{y} = 3$.The ratio of the areas is $r = \frac{	ext{Area}(	riangle AMN)}{	ext{Area}(	riangle ABC)} = xy$.From $\frac{1}{x} + \frac{1}{y} = 3$,we have $y = \frac{x}{3x-1}$.Thus $r = \frac{x^2}{3x-1}$.Since $M$ and $N$ are in the interior of the segments,$0 For $y  1/2$.Also,the minimum value of $r$ occurs when $x=y=2/3$,giving $r = (2/3)(2/3) = 4/9$.As $x 	o 1/2$,$y 	o 1$,so $r 	o 1/2$.Thus,$4/9 \leq r < 1/2$.

In a $\triangle ABC$,let $G$ denote its centroid and let $M, N$ be points in the interiors of the segments $AB, AC$,respectively,such that $M, G, N$ are collinear. If $r$ denotes the ratio of the area of $\triangle AMN$ to the area of $\triangle ABC$,then

Similar Questions

If the position vectors of the vertices of a triangle are $2 \hat{i}-\hat{j}+\hat{k}$,$\hat{i}-3 \hat{j}-5 \hat{k}$,and $3 \hat{i}-4 \hat{j}-4 \hat{k}$,then the triangle is

If the vectors $\hat{i}-2x\hat{j}-3y\hat{k}$ and $\hat{i}+3x\hat{j}+2y\hat{k}$ are orthogonal to each other,then the locus of the point $(x, y)$ is

If $\bar{a}=(2 \hat{i}+2 \hat{j}+3 \hat{k})$,$\bar{b}=(-\hat{i}+2 \hat{j}+\hat{k})$ and $\bar{c}=(3 \hat{i}+\hat{j})$ such that $(\bar{a}+\lambda \bar{b})$ is perpendicular to $\bar{c}$,then the value of $\lambda$ is

If $a=\hat{i}+\hat{j}+t \hat{k}$ and $b=\hat{i}+2 \hat{j}+3 \hat{k}$, then the values of $t$ for which $(a+b)$ and $(a-b)$ are perpendicular are:

Let $\vec{u} = 2\hat{i} + 3\hat{j} + \hat{k}$,$\vec{v} = -3\hat{i} + 2\hat{j}$ and $\vec{w} = \hat{i} - \hat{j} + 4\hat{k}$. Then which of the following statements is true?

Vedclass Test Series

Exam Paper Generator

Online Exam Module