Locus

Q. What is the locus of a point (x, y, z) for which y = 0; z = 0?

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.

The locus of center of the circle which cuts the circles $x^2+y^2+2g_{1}x+2f_{1}y+c_1=0$ and $x^2+y^2+2g_{2}x+2f_{2}y+c_2=0$ orthogonally is

1. The radical axis of the given circle

2. A conic

3. An ellipse

4. Another circle

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. an ellipse with eccentricity $\frac{\sqrt{3}}{4}$
2. a circle of radius 2 units
3. a parabola with focus as (2, 3)
4. an ellipse with length of latusrectum 2 units

Q. What are the direction ratios of negative z axis?

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. $x+y=6$
3. $x+y=2a$
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. $px+qy=40$ is a chord of minimum length of circle $(x-10{)}^2+(y-20{)}^2=729.$ If the chord passes through $(5,15),$ then $p^{2013}+q^{2013}$ is equal to

1. $0$
2. $2$
3. $2^{2013}$
4. $2^{2014}$

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. Q. The number of lines passing through origin and situated at a distance 1 unit from (0, 0) is ?
Can x-axis & y-axis be the two lines satisfying the given condition ? If not, why?

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. Consider two circles $S_1=0$ and $S_2=0$, each of radius $1\text{unit}$ touching internally the sides of $△OAB$ and $△ABC$ respectively. If $O\equiv (0,0),A\equiv (0,4)$ and $B,C$ are the points on positive $x-$axis such that $OB<OC$, then the length of tangent from $A$ to the circle $S_2=0$ is

1. $\frac{9}{4}$ units
2. $9$ units
3. $\frac{7}{2}$ units
4. $\frac{9}{2}$ units

Q.

A bottle of water costs $\$1.50$.

Explain how you would make a graph of four ordered pairs to model the total cost of the water in terms of the number of bottles.

Q. Two circles each of radius $5$ units touch each other at the point $(1,2)$. If the equation of their common tangent is $4x+3y=10$, and $C_1(\alpha ,\beta )$ and $C_2(\gamma ,\delta ),C_1\ne C_2$ are their centres, then $|(\alpha +\beta )(\gamma +\delta )|$ is equal to

Q. A circle of radius ‘r’ is concentric with the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ .Then inclination of common tangent with major axis is _______(b<r<a)

1. 
2. 
3. 
4. 

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

Q. The minimum area of the triangle formed by any tangent to the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ with the coordinate axes is

1. $a^2+b^2$
2. $\frac{(a+b{)}^2}{2}$
3. $ab$
4. $\frac{(a-b{)}^2}{2}$

Q. Three circles touch one another externally. If the tangents at their points of contact meet at a point whose distance from a point of contact is $4$, then the ratio of product of radii to the sum of the radii of the circles, is

1. $2:1$
2. $4:1$
3. $8:1$
4. $16:1$

Q. A circle $C$ of unit radius passing through the centers of the circles $C_1:x^2+y^2=1$ and $C_2:x^2+y^2=2x$ and having center on first quadrant. If $R,$ $R_1$ and $R_2$ are the radical axis of $C_1\&C_2;$ $C\&C_2;$ and $C\&C_1$ respectively, then which of the following is/are correct?

1. $R_2:x+\sqrt{3}y=1$
2. $\text{Radical Center}:(\frac{1}{2},\frac{1}{2\sqrt{3}})$
3. $R,R_1\text{and}{R}_2\text{are concurrent}$
4. $R_2:x-\sqrt{3}y=1$

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. Why is locus defined by using h and f and then replaced with x and y ?

Q. The locus of the mid point of the chord of the circle $x^2+y^2-2x-2y-2=0$ which makes an angle of ${120}^0$ at the centre is.

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

Q. The line $2x+3y+4=0$ cut the circle $x^2+y^2+ax+by+c=0$ at $P$ and $Q$. The line $x-3y+2=0$ cut the circle $x^2+y^2+a^{\prime }x+b^{\prime }y+c^{\prime }$ at $R$ and $S$. If $P,Q,R$ and $S$ are concyclic and value of $\left|\begin{array}{ccc}a-a^{\prime }& b-b^{\prime }& c-c^{\prime }\\ 2& 3& 4\\ 1& -3& 2\end{array}\right|=k\left(abc\right)\left(a^{\prime }{b}^{\prime }{c}^{\prime }\right)$, then $k=$

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. Write the distance of the point (3, −5, 12) from X-axis?

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 straight rod AB of fixed length c moves so that its extremities A , B lie on two fixed mutually perpendicular lines . Prove that the locus of the circumcenter of triangle OAB is a circle , where O is the the point of intersection of mutually perpendicular straight lines

Q.

A variable circle passes through the fixed point (2, 0) and touches the y-axis then the locus of its centre is

1. a parabola

2. a circle

3. an ellipse

4. a hyperbola

Q. $$\begin{array}{cc}\text{List - I}& \text{List - II}\\ \begin{array}{cc}& \text{(I) A circle of radius}r\text{passes through the origin having center}\\ & \text{on}y=x\text{. If it intersects the circle}{x}^2+y^2-10x-6y+2=0\\ & \text{orthogonally, then the value of}4\sqrt{2}r\text{is}\end{array}& \text{(P)}0\\ \begin{array}{cc}& \text{(II) If a pair of tangent is drawn from point}P(-5\sqrt{2},-5\sqrt{2})\text{to}\\ & \text{the circle}{x}^2+y^2=25\text{then the angle subtended by the pair of}\\ & \text{tangents to the given circle is}\frac{\pi }{p}\text{then}p\text{is}\end{array}& \text{(Q)}1\\ \begin{array}{cc}& \text{(III) Let A (-4, 6) and B be a variable point on the line}|x|=6.\\ & \text{If the distance between}A\text{and}B\text{is atmost}4,\text{then the number}\\ & \text{of integral values of B is}\end{array}& \text{(R)}3\\ \begin{array}{cc}& \text{(IV) The value of}\\ & \underset{n\to \infty }{lim}n\sqrt{(\frac{1}{n}-\tan \frac{1}{n})\sqrt{(\frac{1}{n}-\tan \frac{1}{n})\sqrt{(\frac{1}{n}-\tan \frac{1}{n})...\infty }}}\\ & \text{is}\end{array}& \text{(S)}5\\ & \text{(T)}6\\ & \text{(U)}7\end{array}$$
Which of the following is the only CORRECT combination?

1. $\left(II\right),\left(R\right)$
2. $\left(III\right),\left(P\right)$
3. $\left(IV\right),\left(Q\right)$
4. $\left(I\right),\left(S\right)$