Locus

Q. 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).

1. $(\frac{1}{3},-\frac{1}{\sqrt{3}})$
   
2. $(\frac{1}{3},\frac{1}{\sqrt{3}})$
3. $(\frac{1}{4},-\frac{1}{2})$
4. $(\frac{1}{4},\frac{1}{2})$
   

Q.

A ladder starts sliding on a wall from its initial position (fig 1) and (fig 2) is an intermediate stage
before it completely hits the ground. Find the locus of the curve traced by the mid-point of the ladder.
(a and b are constants).

1. 
2. 
3. 
4. 

Q.

A ladder starts sliding on a wall from its initial position (fig 1) and (fig 2) is an intermediate stage
before it completely hits the ground. Find the locus of the curve traced by the mid-point of the ladder.
(a and b are constants).

1. 
2. 
3. 
4. 

Q. A variable chord of circle $x^2+y^2=4$ is drawn from the point $P(3,5)$ meeting the circle at the points $A$ and
$B$. A point $Q$ is taken on this chord such that $2PQ=PA+PB$. Locus of ${}^{\prime \prime }{Q}^{\prime \prime }$ is

1. $x^2+y^2+3x+4y=0$
2. $x^2+y^2=16$
3. $x^2+y^2=36$
4. $x^2+y^2-3x-5y=0$

Q. 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).

1. $(\frac{1}{3},\frac{1}{\sqrt{3}})$
2. $(\frac{1}{4},\frac{1}{2})$
   
3. $(\frac{1}{3},-\frac{1}{\sqrt{3}})$
   
4. $(\frac{1}{4},-\frac{1}{2})$

Q. Locus of the centre of the circle which always passes through the fixed point $(a,0)$ and $(-a,0)$, where $a\ne 0$, is

1. $x=1$
2. $x+y=2a$
3. $x+y=6$
4. $x=0$

Q. A variable circle passes through the point $P(1,2)$ and touches the $x-$axis. The locus of the other end of the diameter through $P$ is

1. $(x-1{)}^2=8y$
2. $(y-1{)}^2=8x$
3. $(x-1{)}^2+8y=0$
4. $(x+1{)}^2=8y$

Q. The circle $x^2+y^2-4x-4y+4=0$ is inscribed in a triangle which has two of its sides along the coordinate axes. If the locus of the circumcentre of the triangle is $x+y-xy+k\sqrt{x^2+y^2}=0$, then the value of $k$ is equal to

1. $2$
2. $3$
3. $-2$
4. $1$

Q. If a variable tangent of the circle $x^2+y^2=1$ intersect the ellipse $x^2+2y^2=4$ at P and Q then the locus of the points of intersection of the tangents at P and Q is

1. A circle of radius 2 units
2. A parabola with fouc as (2, 3)
3. An ellipse with eccentricity
4. An ellipse with length of latus rectrum is 2 units

Q. If one axis of varying standard hyperbola be fixed in magnitude and position, then the locus of the point of contact of tangents drawn to it from a fixed point $(0,c)$ on the other axis is

1. $y^2=\frac{a^2}{c}(x-c)$
   
2. $x^2=-\frac{a^2}{c}(y-c)$
3. $x^2=\frac{a^2}{c}(y+c)$
   
4. $y^2=-\frac{a^2}{c}(x-c)$