Tangent

Q.

If $\left|\frac{\left(z+i\right)}{\left(z-i\right)}\right|=\sqrt{3}$, then the radius of the circle is

1. $\frac{2}{\sqrt{21}}$

2. $\frac{1}{\sqrt{21}}$

3. $\sqrt{3}$

4. $\sqrt{21}$

Q.

The perpendicular distance of center from tangent of the circle is equal to its radius.

1. True

2. False

Q. In the following question, four figures are given in which first two are related to each other in some manner. In the same manner, last two figures should also be related. Which would be the correct alternative for the fourth figure?

1. 
   
2. 
   
   
3. 
   
4. 
   

Q. If the circles $x^2+y^2+(3+\sin \beta )x+\left(2\cos \alpha \right)y=0$ and $x^2+y^2+\left(2\cos \alpha \right)x+2cy=0$ touch each other, then the maximum value of $c$ is

Q. If the lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to a circle, then the radius of the circle is

1. 
2. 
3. 
4. 

Q. If the straight line y = mx + c touches the circle $x^2+y^2-4y=0,$ then the value of c will be

1. $1+\sqrt{1+m^2}$
2. $2(1+\sqrt{1+m^2})$
3. $1-\sqrt{m^2+1}$
4. $2+\sqrt{1+m^2}$

Q. A circle is tangent to the x and y axes in the first quadrant at the points $P$ and $Q$ respectively. $BC$ and $AD$ are parallel tangents to the circle with slope $-1$. If the points $A$ and $B$ are on the y- axis while $C$ and $D$ are on the x-axis and the area of the quadrilateral $ABCD$ is $900\sqrt{2}$ sq. units, then the radius of the circle is

1. $10$
2. $12$
3. $15$
4. $20$

Q.

Circle drawn through the point $(2,0)$ to cut intercept of length $‘5^{\prime }$ units on the x-axis. If its centre lie in the first quadrant then the equation of family of such circles is

1. $x^2+y^2-9x+2ky+14=0,k>0$

2. $x^2+y^2-9x-2ky+14=0,k>0$

3. $x^2+y^2-9y+2ky+14=0,k>0$

4. $3x^2+3y^2+27x+2ky+42=0,k>0$

Q. Let $W_1$ and $W_2$ denote the circles $x^2+y^2+10x-24y-87=0$ and $x^2+y^2-10x-24y+153=0$ respectively. Let $m$ be the smallest positive value of $a$ for which the line $y=ax$ contains the centre of a circle that is externally tangent to $W_2$ and internally tangent to $W_1.$ If $m^2=\frac{p}{q},$ where $p$ and $q$ are co-prime, then the value of $(p+q)$ is

Q. Tangents $PA$ and $PB$ are drawn to the circle $(x-4{)}^2+(y-5{)}^2=4$ from the point $P$ on the curve $y=\sin x$, where $A$ and $B$ lie on the circle. Consider the function y = $f\left(x\right)$ represented by the locus of the centre of the circumcircle of triangle $PAB$, then
Range of $y=f\left(x\right)$ is

1. $[-2,1]$
2. $[-1,4]$
3. $[0,2]$
4. $[2,3]$

Q.

A plane passes through (1, -2, 1) and is perpendicular to two planes $2x-2y+z=0$ and $x-y+2z=4,$ then the distance of the plane form the point (1, 2, 2) is

1. 0

2. $\sqrt{2}$

3. 1

4. $2\sqrt{2}$

Q.

Find the coordinates of the point on the curve $\sqrt{x}+\sqrt{y}=4$ at which tangent is equally inclined to the axes.

Q. The two tangents to a circle from an external point are always

1. Equal

2. Perpendicular to each other

3. Parallel to each other
4. None of these

Q. If the line lx + my = 1 be a tangent to the circle $x^2+y^2=a^2,$ then the locus of the point (l, m) is

1. A straight line

2. An ellipse

3. A Circle

4. A parabola

Q. Consider the relation $4l^2-5m^2+6l+1=0$, where $l,m\in R$, then the line $lx+my+1=0$ touches a fixed circle whose centre and radius of circle are

1. $(2,0),3$
2. $(-3,0),\sqrt{3}$
3. $(3,0),\sqrt{5}$
4. $(-2,0),3$

Q. The equation of circle with centre (1, 2) and tangent x + y - 5 = 0 is

1. $x^2+y^2-2x-4y+3=0$
2. $x^2+y^2+2x-4y+6=0$
3. $x^2+y^2-2x-4y+8=0$
4. $x^2+y^2-2x+4y+8=0$

Q. The shortest distance between the line $y=x-5$ and the parabola $y=x^2+3x+6$ is

1. $\sqrt{2}\text{units}$
2. $2\sqrt{2}\text{units}$
3. $3\sqrt{2}\text{units}$
4. $5\sqrt{2}\text{units}$

Q. Let the abscissae of the two points $P$ and $Q$ be the roots of $2x^2–rx+p=0$ and the ordinates of $P$ and $Q$ be the roots of $x^2–sx–q=0$. If the equation of the circle described on $PQ$ as diameter is $2(x^2+y^2)–11x–14y–22=0$, then $2r+s–2q+p$ is equal to .

Q.

The foot of the perpendicular of center on a tangent and the point of contact of the tangent are same.

1. True

2. False

Q.

The set of values of 'c' so that the equations $y=\left|x\right|+cand{x}^2+y^2-8\left|x\right|-9=0$ have no solution is

1. $(\infty ,-3)\cup (3,\infty )$
   

2. $(-3,3)$
   

3. $(-\infty ,5\sqrt{2})\cup \left(5\sqrt{(}2\right),\infty )$
   

4. $5\sqrt{2}-4,\infty$

Q. The equation of the circle lying in first quadrant, which touches both the coordinates axes and the distance of it's centre from origin is $2$ units is

1. $x^2+y^2-2\sqrt{2}x-2y+2=0$
2. $x^2+y^2-\sqrt{2}x-2\sqrt{2}y+2=0$
3. $x^2+y^2-2x-2y+2=0$
4. $x^2+y^2-2\sqrt{2}x-2\sqrt{2}y+2=0$

Q. If a circle is inscribed in a square of side $10$ , so that the circle touches the four sides of the square internally then radius of the circle is

1. $10$
2. $5\sqrt{2}$
3. $10\sqrt{2}$
4. $5$

Q.

The equation of the circle is $x^2+y^2-12x+6y+20=0$.

What is the radius of the circle?

Enter your answer in the box. $r=$ ___ units.

Q.

The perpendicular distance of center from tangent of the circle is equal to its radius.

1. True

2. False

Q. In the figure given below, three dots are placed that represent some common regions to circle, hexagon and triangle.
Choose the correct alternative figure that contains the similar common region to that represented by dots in the given figure.

1. 
   
2. 
   
3. 
   
4. 
   

Q. If the straight line y = mx + c touches the circle $x^2+y^2-4y=0,$ then the value of c will be

1. $1+\sqrt{1+m^2}$
2. $1-\sqrt{m^2+1}$
3. $2(1+\sqrt{1+m^2})$
4. $2+\sqrt{1+m^2}$

Q. Consider the circles
$S_1:x^2+y^2-12y+35=0$
$S_2:x^2+y^2-4x-4y+4=0$
$S_1:x^2+y^2-9x=0$
Which of the following statements is correct?

1. The centres of these three circles form a right triangle.
2. The length of the chord intercepted on the line $y=x$ by $S_3$ is $3\sqrt{2}.$
3. Equation of the tangent on circle $S_3=0$ at the origin is $y=0$
4. Point $(4,1)$ lies outside $S_1$ but inside $S_2$ and $S_3$.

Q. If the lines 3x - 4y + 4 = 0 and 6x - 8y - 7 = 0 are tangents to a circle, then the radius of the circle is

1. 3/2
2. 3/4
3. 1/10
4. 1/20

Q. The shortest distance from the origin to a variable point on the sphere $(x-2{)}^2+(y-3{)}^2+(z-6{)}^2=1$ is

1. $5$
2. $6$
3. $7$
4. $8$

Q. Consider the relation $4l^2-5m^2+6l+1=0$, where $l,m\in R$, then the line $lx+my+1=0$ touches a fixed circle whose centre and radius of circle are

1. $(-3,0),\sqrt{3}$
2. $(2,0),3$
3. $(3,0),\sqrt{5}$
4. $(-2,0),3$