Dr. SK Goyal Solutions for Chapter: Coordinate Systems and Coordinates, Exercise 5: EXERCISE ON LEVEL-I

$A$ and $B$ are two points whose co-ordinates are $\left(a{t}^2,2at\right)$ and $\left(\frac{a}{t^2},-\frac{2a}{t}\right)$ and $S$ is the point $(a, 0)$. Prove that $\frac{1}{SA}+\frac{1}{SB}=\frac{1}{a}$.

$A\left(a_1,b_1\right),B\left(a_2,b_2\right),C\left(a_3,b_3\right)$ are the vertices of a $∆ABC$. The side $AB$ is divided by the point $D$ in the ratio $\lambda :\mu$ and then the line segment $DC$ is divided by the point $E$ in the ratio $v:\lambda +\mu$. Find the co-ordinates of $E$.

If the area of the quadrilateral whose angular points taken in order are $(1,2),(-5,6),(7,-4)$ and $(\lambda ,-2)$ be zero, show that $\lambda -3=0$.

If $\tan \alpha ,\tan \beta ,\tan \gamma$are the roots of the equation $x^3-3a{x}^2+3bx-1=0$. Find the centroid of the triangle whose vertices are $(\tan \alpha ,\cot \alpha ),(\tan \beta ,\cot \beta ),(\tan \gamma ,\cot \gamma ).$

$A\left(2,3\right)$ is a fixed point and $Q\left(3\cos \theta , 2\sin \theta \right)$ a variable point. If $P$ divides $AQ$ internally in the ratio $3:1,$ find the locus of $P$.

$A$ is the point $\left(4,0\right)$ and $M$ is the foot of the perpendicular drawn from a point $P$ to the $y$ -axis. If P moves such that the distances $PA$ and $PM$ remain equal, find the locus of $P.$

In any triangle $ABC,$ prove that
$A{B}^2+A{C}^2=2\left(A{D}^2+D{C}^2\right)$
where $D$ is the middle point of $BC$.

$A,B,C$ are the points $(-1,5),(3,1)$ and $(5,7)$ respectively. $D,E,F$ are the middle points of $BC,CA$ and $AB$ respectively. Prove that $∆ABC=4∆DEF$.