Director Circle

Q. The equation of director circle of the circle $x^2+y^2=a^2$ is

1. None of these
2. $x^2+y^2=4a^2$
3. $x^2+y^2=\sqrt{2}{a}^2$
4. $x^2+y^2-2a^2=0$

Q. The equation of the director circle of the circle $(x-2{)}^2+(y+3{)}^2=16$ is .

1. $x^2+y^2-4x+6y=45$
2. $x^2+y^2-4x+6y=19$
3. $x^2+y^2-4x+6y=51$
4. $x^2+y^2-4x+6y=77$

Q. If two tangents drawn from a point $P$ to the parabola $y^2=16(x–3)$ are at right angles, then the locus of point $P$ is

1. $x+2=0$
2. $x+4=0$
3. $x+1=0$
4. $x+3=0$

Q. The number of points on the curve $y=||1-e^x|-2|$ from which mutually perpendicular tangents can be drawn to the parabola $x^2=-4y$ is

Q. Tangents are drawn to the circle $x^2+y^2=50$ from a point $P$ lying on the $x-$axis. These tangents meet the $y-$axis at points $P_1$ and $P_2$. Possible coordinates of $P$ so that area of $△P{P}_1{P}_2$ is minimum, are

1. $(10,0)$
2. $(10\sqrt{2},0)$
3. $(-10,0)$
4. $(-10\sqrt{2},0)$

Q.

Find the real numbers x and y if (x – iy) (3 + 5i) is the conjugate of –6 – 24i.

Q.

Tangent at any point on the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ cut the axis at A and B respectively. If the rectangle OAPB (where O is origin) is completed then locus of point P is given by

1. 

2. 

3. 

4. None of these

Q. If the angle between the tangents drawn to the circle $x^2+y^2+2gx+2fy+c=0$ $(c>0)$ from the origin is $\frac{\pi }{2},$ then

1. $g^2+f^2=3c$
2. $g^2+f^2=2c$
3. $g^2+f^2=c$
4. $g^2+f^2=4c$

Q. The points of contact Q and R tangent from the point $P(2,3)$ on the parabola $y^2=4x$ are

1. $(1,2),(4,4)$
2. $(9,6),(\frac{1}{4},1)$
3. $(9,6),(1,2)$
4. $(4,4),(9,6)$

Q. If the radius of the circumcircle of the triangle $TPQ$, where $PQ$ is chord of contact corresponding to point $T$ with respect to circle $x^2+y^2-2x+4y-11=0$, is $6$ units, then minimum distance of $T$ from the director circle of the given circle is

1. $6$
2. $12$
3. $6\sqrt{2}$
4. $12-4\sqrt{2}$

Q. The equation of the circle $C_1$ is $x^2+y^2=4$. The locus of the intersection of orthogonal tangents to the circle is the curve $C_2$ and the locus of the intersection of perpendicular tangents to the curve $C_2$ is the curve $C_3$. Then

1. $C_3$ is a circle
2. the area enclosed by the curve $C_3$ is $8\pi$
3. $C_2$ and $C_3$ are circles with same centre
4. radius of $C_3$ is twice the radius of $C_2$

Q. Let $f(x,y)=0$ be the equation of a circle such that $f(0,y)=0$ has equal real roots and $f(x,0)=0$ has two distinct real roots. Let $g(x,y)=0$ be the locus of points ${}^{\prime }{P}^{\prime }$ from where tangents to circle $f(x,y)=0$ make angle $\frac{\pi }{3}$ between them and $g(x,y)=x^2+y^2–5x–4y+c,c\in R$. Let $Q$ be a point from where tangents drawn to circle $g(x,y)=0$ are mutually perpendicular. If $A,B$ are the points of contact of tangents drawn from $Q$ to circle $g(x,y)=0$, then area of triangle $QAB$ is

1. $\frac{25}{12}$
2. $\frac{25}{2}$
3. $\frac{25}{8}$
4. $\frac{25}{4}$

Q. The minimum distance between any two points $P_1$ and $P_2$ while considering point $P_1$ on one circle and point $P_2$ on the other circle for the given circle's equations $x^2+y^2-10x-10y+41=0,x^2+y^2-24x-10y+160=0$

Q.

The difference of the radii of the two circles with centre (4, 3) and touching the circle $x^2+y^2=1$, is

1. 0

2. 2

3. 4

4. Not fixed

Q.

The equation of the circle passing through the points (4, 1), (6, 5) whose centre lies on the line 4x+y-16=0 is

1. x ²+y²-6x-8y+15=0

2. x ²+y²-10x-6y+29=0

3. x ²+y²-x-13=0

4. x ²+y²+4y-12=0

Q. Let $P(x,y)$ be a variable point such that $|\sqrt{(x-2{)}^2+(y-3{)}^2}-\sqrt{(x-6{)}^2+(y-6{)}^2}|=4$, then

1. The locus of $P$ is a hyperbola with eccentricity $\frac{5}{4}$
2. The locus of $P$ is a hyperbola with eccentricity $\frac{5}{3}$
3. The locus of the intersection of perpendicular tangents to the locus of $P$ is ${(x-4)}^2+{(y-\frac{9}{2})}^2=\frac{5}{4}$
4. The locus of the intersection of perpendicular tangents to the locus of $P$ is ${(x-4)}^2+{(y-\frac{9}{2})}^2=\frac{7}{4}$

Q.

If the ratio of the lengths of tangents from a point to the circles $x^2+y^2+4x+3$ = 0, $x^2+y^2-6x+5$ = 0 is 1:2 then the locus of P is a circle whose centre is

1. (-11/3 , 0 )

2. (-22/3 , 0 )

3. (-11 , 6 )

4. (-11/5 , 0 )

Q. A circle is concentric with the circle $x^2+y^2-6x+12y+15=0$ and has area double of its area. The equation of the circle is

1. 
2. 
3. 
4. None of these

Q. Let $f(x,y)=0$ be the equation of a circle such that $f(0,y)=0$ has equal real roots and $f(x,0)=0$ has two distinct real roots. Let $g(x,y)=0$ be the locus of points ${}^{\prime }{P}^{\prime }$ from where tangents to circle $f(x,y)=0$ make angle $\frac{\pi }{3}$ between them and $g(x,y)=x^2+y^2–5x–4y+c,c\in R$. Let $Q$ be a point from where tangents drawn to circle $g(x,y)=0$ are mutually perpendicular. If $A,B$ are the points of contact of tangents drawn from $Q$ to circle $g(x,y)=0$, then area of triangle $QAB$ is

1. $\frac{25}{12}$
2. $\frac{25}{8}$
3. $\frac{25}{4}$
4. $\frac{25}{2}$

Q. $TP$ and $TQ$ are tangents to the parabola $y^2=4ax$ at points $P$ and $Q$. If the chord $PQ$ passes through a fixed point $(–a,b)$, then the locus of $T$ is

1. $by=2a(x–a)$
2. $ax=2b(y–b)$
3. $bx=2a(y–a)$
4. $ay=2b(x–b)$

Q.

Find the equation of director circle of a circle whose equation is $x^2+y^2-4x+6y+12=0$

1. $x^2+y^2-4x+6y+11=0$

2. $x^2+y^2-4x+6y+12=0$

3. $x^2+y^2-2x+3y+11=0$

4. $x^2+y^2-4x+6y+1=0$

Q.

Find the equation of director circle of the circle $(x-1{)}^2+(y-2{)}^2=4$

1. $(x-1{)}^2+(y-2{)}^2=4\sqrt{2}$

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

3. $(x+1{)}^2+(y+2{)}^2=4\sqrt{2}$

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

Q.

AB is a chord of the parabola $y^2=4ax$ with vertex at A. BC is drawn perpendicular to AB meeting the axis at C. The projection of BC on the x-axis is-

1. a

2. 2a

3. 4a

4. 8a

Q. Find the center and radius of the circle whose equation is given by $x^2+y^2+4y=0$

Q. Find the equations of the tangent and normal to the parabola $y^2=8x$ at $(2,4)$.

Q. Let PQ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at P and Q meet at a point lying on the line $y=2x+a,a>0$.
Length of chord PQ is

1. 2a
2. 7a
3. 5a
4. 3a

Q. Match $\text{List I}$ with the $\text{List II}$ and select the correct answer using the code given below the lists :

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Radius of the largest circle which passes through the focus of the parabola}{y}^2=4x\text{and is completely}& \text{(P)}& 16\\ & \text{contained in it, is}& & \\ \text{(B)}& \text{If the shortest distance between the curves}{y}^2=4x\text{and}{y}^2=2x–6\text{is}d\text{, then}{d}^2\text{is}& \text{(Q)}& 5\\ \text{(C)}& \text{Let}AB\text{be a focal chord of}{y}^2=12x\text{with focus}S.\text{The harmonic mean of lengths of segments}AS& \text{(R)}& 6\\ & \text{and}BS\text{is}& & \\ \text{(D)}& \text{Tangents drawn from}P\text{meet the parabola}{y}^2=16x\text{at}A\text{and}B.\text{If these two tangents are}& \text{(S)}& 4\\ & \text{perpendicular, then the least value of}\sqrt{AB}\text{is}& & \end{array}$


Which of the following is CORRECT ?

1. $\left(C\right)\to \left(R\right),\left(D\right)\to \left(S\right)$
2. $\left(C\right)\to \left(P\right),\left(S\right)\to \left(Q\right)$
3. $\left(C\right)\to \left(Q\right),\left(D\right)\to \left(Q\right)$
4. $\left(C\right)\to \left(R\right),\left(D\right)\to \left(Q\right)$

Q. Assertion :The maximum value of $(\sqrt{-3+4x-x^2}+4{)}^2+(x-5{)}^2$ (where $1\le x\le 3)$ is $36$. Reason: The maximum distance between the point $(5,-4)$ and the point on the circle $(x-2{)}^2+y^2=1$ is $6$.

1. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
2. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
3. Assertion is correct but Reason is incorrect
4. Both Assertion and Reason are incorrect

Q.

Which of the following statements is/are correct?

1. The locus of the point of intersection of two perpendicular tangents is called director circle of the given circle.

2. The director circle of a circle is concentric circle.

3. The radius of director circle of a circle is equal to the radius of original circle.C

1. Only 1 & 2

2. Only 1

3. Only 1 & 3

4. All 1, 2 and 3

Q. If the tangent at the point $P(2,4)$ to the parabola $y^2=8x$ meets the parabola $y^2=8x+5$ at $Q$ and $R$, then the midpoint of $QR$ is

1. $(2,4)$
2. $(4,2)$
3. $(7,9)$
4. none of these