Point in Cartesian Plane

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 $R$.

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 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 $R$.

The points $\left(1,3\right)$ and $\left(5,1\right)$ are two opposite vertices of a rectangle. The other two vertices lie on the line $y=2x+c$, then $c$ is -

If the point $P\left(3,5\right)$ divides the line joining $A\left(1,3\right)$ & $B\left(7,9\right)$ and harmonic conjugate of $P$ w.r.t.$A$ & $B$ is $Q\left(a,b\right)$, then $a+b$ is

Coordinates of a point which is at $3$ units distance from the point $\left(1,-3\right)$ on the line $2x+3y+7=0$ is/are-

If one vertex of an equilateral triangle of side '$a$' lies at the origin and the other lies on the line $x-\sqrt{3}y=0$, then the co-ordinates of the third vertex are-

The line $x+y=p$ meets the axis of $x$ and $y$ 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 $3/8^{\mathrm{th}}$ of the area of the triangle $OAB$, then $\frac{AQ}{BQ}$ is equal to-

The points $\left(0,\frac{8}{3}\right)$, $\left(1,3\right)$ and $\left(82,30\right)$ are:

A stick of length $10$ units rests against the floor and a wall of a room. If the stick begins to slide on the floor then the locus of its middle point is-

The points with the co-ordinates $\left(2a,3a\right),\left(3b,2b\right)$ and $\left(c,c\right)$ are collinear $\left(a,b,c\ne 0\right)$:

The point $A$ divides the join of the points $\left(-5,1\right)$ and $\left(3,5\right)$ in the ratio $k:1$ and coordinates of points $B$ and $C$ are $\left(1,5\right)$ and $\left(7,-2\right)$ respectively. If the area of $\Delta ABC$ be $2$ units, then $k$ equals-

The ratio in which the line joining the points $\left(3,-4\right)$ and $\left(-5,6\right)$ is divided by $x$-axis-

If $\left(3,-4\right)$ and $\left(-6,5\right)$ are the extremities of a diagonal of a parallelogram and $\left(2,1\right)$ is the third vertex, then its fourth vertex is-

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.

Circle are drawn which are orthogonal to both the circles $S\equiv x^2+y^2-16=0$ and $S^{\text{'}}\equiv x^2+y^2-8x-12y+16=0$. If tangents are drawn from the centre of the variable circles to $S$. Then find the locus of the mid point of the chord of contact of these tangents.

Show that the locus of the centres of a circle which cuts two given circles orthogonally is a straight line & hence deduce the locus of the centre of the circles which cut the circles $x^2+y^2+4x-6y+9=0$ & $x^2+y^2-5x+4y+2=0$ orthogonally.

A triangle has two of its sides along the coordinate axes, its third side touches the circle $x^2+y^2-2ax-2ay+a^2=0$. Prove that the locus of the circumcentre of the triangle is: $a^2-2a\left(x+y\right)+2xy=0$.

A variable circle passes through the point $A\left(a,b\right)$ and touches the $x$-axis. Show that the locus of the other end of the diameter through $A$ is ${\left(x-a\right)}^2=4by$.

If $A(\cos \alpha ,\sin \alpha ), B(\sin \alpha ,-\cos \alpha ), C(1, 2)$ are the vertices of a $\Delta ABC,$ then as $\alpha$ varies, the locus of its centroid is: