Embibe Experts Solutions for Chapter: Point and Straight Line, Exercise 4: Exercise-4

Let in $∆PAB, A$ is $\left(0, 0\right), B$ is $\left(a, 0\right)$ and $P$ is a variable point such that $\angle PBA$ is equal to three times $\angle PAB$. Then find the locus of $P.$

Through the origin $O$, a straight line is drawn to cut the lines $y=m_{1}x+c_1$ and $y=m_{2}x+c_2$ at $Q$ and $R$, respectively. Find the locus of the point $P$ on this variable line, such that OP, is the geometric mean between $OQ$ and $OR$.

The straight lines $\left(A^2-3B^2\right)x^2+8ABxy+\left(B^2-3A^2\right)y^2=0$ form a $\Delta$ with the line $Ax+By+C=0$, then prove that
(i) Area of the $\Delta$ is $\frac{C^2}{\sqrt{3}\left(A^2+B^2\right)}$
(ii) $\Delta$ is equilateral
(iii) The orthocentre of $\Delta$ does not lie on one of its vertex

Find the acute angle between two straight lines passing through the point $M(-6,-8)$ and the points in which the line segment $2x+y+10=0$ enclosed between the coordinate axes is divided in the ratio $1: 2: 2$ in the direction from the point of its intersection with the $x$-axis to the point of intersection with the $y$-axis.

Let $A$ lies on $3x-4y+1=0$. $B$ lies on $4x+3y-7=0$ and $C$ is $\left(-2, 5\right)$. If $ABCD$ is rhombus, then find locus of $D$

Let $D$ is point on line ${\ell }_1:x+y-2=0$ and $s\left(3, 3\right)$ is fixed point. ${\ell }_2$ is the perpendicular to $DS$ and passing through $S.$ If $M$ is another point on line ${\ell }_1$ (other than $D$), then find locus of point of intersection of ${\ell }_2$ and angle bisector of $\angle MDS.$

A variable line cuts the line $2y=x-2$ and $2y=-x+2$ in points $A$ and $B$ respectively. If $A$ lies in first quadrant, $B$ lies in $4^{\mathrm{th}}$ quadrant and area of $∆AOB$ is $4$ sq. units, then find locus of mid point of $AB$.

A variable line cuts the line $2y=x-2$ and $2y=-x+2$ in points $A$ and $B$ respectively. If $A$ lies in first quadrant, $B$ lies in $4^{\mathrm{th}}$ quadrant and area of $∆AOB$ is $4$ sq. units, then find locus of centroid of $∆AOB$