Locus

Let $RS$ be the diameter of the circle $x^2+y^2=1$, where $S$ is the point $(1,0)$. Let $P$ be a variable point (other than $R$ and $S$) on the circle and tangents to the circle at $S$ and $P$ meet at the point $Q$. The normal to the circle at $P$ intersects a line drawn through $Q$ parallel to $RS$ at point $E$. Then the locus of $E$ passes through the point(s)

If a variable line drawn through the intersection of the lines $\dfrac x3+\dfrac y4=1$ and $\dfrac x4+\dfrac y3=1$, meets the coordinate axes at A and B, (A ≠ B), then the locus of the midpoint of AB is :

Let the curve $C$ be the mirror image of the parabola $y^2=4x$ with respect to the line $x+y+4=0$. If $A$ and $B$ are the points of intersection of $C$ with the line $y=-5$, then the distance between $A$ and $B$ is

The number of points, having both co-ordinates as integers, that lie in the interior of the triangle with vertices (0, 0), (0, 41) and (41, 0), is :

Let O be the vertex and Q be any point on the parabola, $x^2=8y$. If the point P divides the line segment OQ internally in the ratio 1 : 3, then the locus of P is :

Let PQ be a double ordinate of the parabola, $y^2=-4x$, where P lies in the second quadrant. If R divides PQ in the ratio 2 : 1, then the locus of R is :

The locus of the foot of perpendicular drawn from the centre of the ellipse $x^2+3y^2=6$ on any tangent to it is :

Let $a$ and $b$ be any two numbers satisfying $\dfrac1{a^2}+\dfrac1{b^2}=\dfrac14$. Then, the foot of perpendicular from the origin on the variable line, $\dfrac xa+\dfrac yb=1$, lies on :

A stair-case of length $l$ rests against a vertical wall and a floor of a room,. Let P be a point on the stair-case, nearer to its end on the wall, that divides its length in the ratio 1 : 2. If the stair-case begins to slide on the floor, then the locus of P is :

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

Let $(x,y)$ be any point on the parabola $y^2=4x$. Let $P$ be the point that divides the line segment from $(0,0)$ to $(x,y)$ in the ratio $1:3$. Then the locus of $P$ is

The equation of the locus of the point whose distances from the point P and the line AB are equal, is

The normal at a point $P$ on the ellipse $x^2+4y^2=16$ meets the $x$-axis at $Q$. If $M$ is the mid point of the line segment $PQ$, then the locus of $M$ intersects the latus rectums of the given ellipse at the points

The locus of the orthocentre of the triangle formed by the lines
$$(1+p)x-py+p(1+p)=0,\qquad(1+q)x-qy+q(1+q)=0,$$
and $y=0$, where $p\ne q$, is

The tangent $PT$ and the normal $PN$ to the parabola $y^2=4ax$ at a point $P$ on it meet its axis at points $T$ and $N$, respectively. The locus of the centroid of the triangle $PTN$ is a parabola whose