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

Find the locus of the mid point of the chord of a circle ${\mathrm{x}}^2+{\mathrm{y}}^2=4$ such that the segment intercepted by the chord on the curve $x^2-2x-2y=0$ subtends a right angle at the origin.

A line cuts the $x$-axis at $A\left(7,0\right)$ and the $y$-axis at $B\left(0,-5\right)$. A variable line $PQ$ is drawn perpendicular to $AB$ cutting the $x$-axis in $P$ and the $y$-axis in $Q$. If $AQ$ and $BP$ intersect at $R$, find the locus of $R$.

Find the equation of the line which bisects the obtuse angle between the lines $x-2y+4=0$ and $4x-3y+2=0$.

Show that all the chords of the curve $3x^2+3y^2-2x+4y=0$ which subtend a right angle at the origin are concurrent. Also find the point of concurrency.

Determine all values of $\alpha$ for which the point $\left(\alpha ,{\alpha }^2\right)$ lies inside the triangle formed by the lines $2x+3y-1=0;x+2y-3=0;5x-6y-1=0$.

Lines $L_1\equiv ax+by+c=0$ and $L_2\equiv lx+my+n=0$ intersect at the point $P$ and makes an angle $\theta$ with each other. Find the equation of a line $L$ different from $L_2$ which passes through $P$ and makes the same angle $\theta$ with $L_1$.

Equation of a line is given by $y+2at=t\left(x-a{t}^2\right)$, $t$ being the parameter. Find the locus of the point intersection of the lines which are at right angles.

A rectangle $PQRS$ has its side $PQ$ parallel to the line $y=mx$ and vertices $P,Q$ and $S$ on the lines $y=a, x=b$ and $x=-b$, respectively. Find the locus of the vertex $R$.