Embibe Experts Solutions for Chapter: Point and Straight Line, Exercise 3: Level 3

The line $x+y=p$ meets the $x$ and $y$ -axes at $A$ and $B$, respectively. A triangle $APQ$ is inscribed in the triangle $OAB, O$ being the origin, with right angle at $Q$. $P$ and $Q$ lie, respectively, on $OB$ and $AB$. If the area of the triangle $APQ$ is ${\left(\frac{3}{8}\right)}^{\mathrm{th}}$ of the area of the triangle $OAB,$ then $\frac{AQ}{BQ}$ is equal to

Let $A_r, r=1,2,3\dots ..$ be points on the number line such that $O{A}_1, O{A}_2, O{A}_3, \dots ..$ are in $\mathrm{G}.\mathrm{P}.$ be a positive proper fraction. Let $M_r$ be the middle point of the line segment $A_r{A}_{r+1}$. Then the value of $\sum _{r=1}^{\infty }O{M}_r$ is equal to

If $ABCD$ is a $2\times 2$ square, $E$ is the midpoint of $\overline{AB}, F$ is the midpoint of $\overline{BC}, \overline{AF}$ and $\overline{DE}$ intersect at $I$ and $BD$ and $AF$ intersect at $H$ and the area of the quadrilateral $BEIH$ is $\frac{k}{2k+1}$, find $k.$

The equation of the straight line in the normal form which is parallel to the lines $x+2y+3=0$ and $x+2y+8=0$ and dividing the distance between these two lines in the ratio $1:2$ internally is

If $P\left(1+\frac{t}{\sqrt{2}},2+\frac{t}{\sqrt{2}}\right)$ be any point on a line, then the range of the values of $t$ for which the point $P$ lies between the parallel lines $x+2y=1$ and $2x+4y=15$ is

${\theta }_1$ and ${\theta }_2$ are the inclination of lines $L_1$ and $L_2$ with $x$-axis. If $L_1$ and $L_2$ pass through $P\left(x_1,y_1\right)$, then equation of one of the angle bisector of these lines is

If the orthocenter of the triangle formed by the lines $2x+3 y-1=0, x+2y+1=0$ and $ax+by-1=0$ lies at origin, then $\frac{1}{a}+\frac{1}{b}=$

Let $C$ be the centroid of the triangle with vertices $(3,-1),(1,3)$ and $(2,4).$ Let $P$ be the point of intersection of the lines $x+3 y-1=0$ and $3 x-y+1=0.$ Then the line passing through the points $C$ and $P$ also passes through the point