Position of a Point with Respect to Circle

Q. Let $T$ be the line passing through the points $P(-2,7)$ and $Q(2,-5)$. Let $F_1$ be the set of all pairs of circles $(S_1,S_2)$ such that $T$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$, and also such that $S_1$ and $S_2$ touch each other at a point, say, $M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1,S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1,1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then, which of the following statement(s) is (are) TRUE?

1. The point $(-2,7)$ lies in $E_1$.
2. The point $(\frac{4}{5},\frac{7}{5})$ does $\text{NOT}$ lie in $E_2$.
3. The point $(\frac{1}{2},1)$ lies in $E_2$.
4. The point $(0,\frac{3}{2})$ does $\text{NOT}$ lie in $E_1$.

Q. The range of parameter $a$ for which the variable line $y=2x+a$ lies between the circles $x^2+y^2-2x-2y+1=0$ and $x^2+y^2-16x-2y+61=0$ without intersecting or touching either circle is

1. $(-15,-1)$
2. $(\sqrt{5}-1,\infty )$
3. $(-\infty ,-2\sqrt{5}-15)$
4. $(2\sqrt{5}-15,-\sqrt{5}-1)$

Q.

If OA and OB be the tangents to the circle $x^2+y^2-6x-8y+21=0$ drawn from the origin O, then AB =

1. 11

2. 12

3. 

4. 

Q. $\frac{x-x_1}{\cos \theta }=\frac{y-y_1}{\sin \theta }=r$, represents:

1. Equation of straight line, if $\theta$ is variable & $r$ is constant
2. Equation of circle, if $r$ is variable & $\theta$ is constant
3. Equation of circle, if $\theta$ is variable & $r$ is constant
4. Equation of straight line, if $r$ is variable & $\theta$ is constant

Q. The nearest point on the circle $x^2+y^2-6x+4y-12=0$ from the point $P(-5,4)$ is $Q(\alpha ,\beta ),$ then the value of $\alpha +\beta$ is

1. $0$
2. $4$
3. $2$
4. $7$

Q. The range of values of $\alpha$ for which the point $(\alpha ,\alpha )$ lies in the interior part of smaller segment of $x^2+y^2=4$ intercepted by the line $3x+4y+7=0,$ is

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

Q. The range of values of $r$, for which the point $(-5+\frac{r}{\sqrt{2}},-3+\frac{r}{\sqrt{2}})$ is an interior point of the major segment of the circle $x^2+y^2=16$ cut-off by the line $x+y=2$, is

1. $(4\sqrt{2}-\sqrt{14},5\sqrt{2})$
2. $(4\sqrt{2}-\sqrt{14},4\sqrt{2}+\sqrt{14})$
3. None of the above
4. $(-\infty ,5\sqrt{2})$

Q. The number of tangents which can be drawn from the point (–1, 2) to the circle $x^2+y^2+2x-4y+4=0$ is

1. 1
2. 2
3. 0
4. 3

Q. If $P(0,0),Q(1,0)$ and $R(\frac{1}{2},\frac{\sqrt{3}}{2})$ are three given points, then the centre of the circle for which the lines $PQ,QR$ and $RP$ are the tangents is

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

Q. If a line makes an angles $\alpha ,\beta ,\gamma$ with positive axes. Then the range of $\sin \alpha \sin \beta +\sin \beta \sin \gamma +\sin \alpha \sin \gamma$ is:

1. $[\frac{-1}{2},1]$
2. $[\frac{1}{2},2]$
3. $[-1,2]$
4. $(-1,2]$

Q. If $d_1$ and $d_2$ are the longest and the shortest distances of the point $P(-7,2)$ from the circle $x^2+y^2-10x-14y-51=0,$ then the value of $d_1^2+d_2^2$ is

Q.

The minimum radius vector of the curve $\frac{a^2}{x^2}+\frac{b^2}{y^2}=1$ is of length

1. $a-b$

2. $\left|a+b\right|$

3. $2a+b$

4. None of these

Q. If number of integral coordinates $(x,y)$ which lie inside to the circle $x^2+y^2=25$ is $n,$ then $\left[\frac{n}{9}\right]$ is
( $[.]$ represents the greatest integer function. )

Q. The set of values of $a$ for which the point$(a-1,a+1)$ lies outside the circle $x^2+y^2=8$ and inside the circle $x^2+y^2-12x+12y-62=0$ is

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

Q. The area of the loop of the curve $y^2=x^4(x+2)$ is [in square units]

1. $\frac{32\sqrt{2}}{105}$
2. $\frac{64\sqrt{2}}{105}$
3. $\frac{128\sqrt{2}}{105}$
4. $\frac{256\sqrt{2}}{105}$

Q. If $(a,0)$ is an endpoint of a diameter of the circle $x^2+y^2=4$, then $x^2-4x-a^2=0$ has

1. exactly one real root in $(-1,0]$
2. exactly one real root in $[2,5]$
3. distinct roots greater than $-1$
4. distinct roots less than $5$

Q. A tower subtends angles $\theta ,2\theta$ and $3\theta$ at three points $A,B,C$ respectively lying on a horizontal line through the foot of tower. Then the ratio $\frac{AB}{BC}$ equals

1. $\frac{\sin 3\theta }{\sin \theta }$
2. $\frac{\sin \theta }{\sin 3\theta }$
3. $\frac{\cos 3\theta }{\cos \theta }$
4. $\frac{\tan \theta }{\tan 3\theta }$

Q. From the point $P(2,1),$ a line of slope $m\in R$ is drawn so as to cut the circle $x^2+y^2=1$ in points $A$ and $B$. If the slope $m$ is varied, then the greatest possible value of $PA+PB$ is

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

Q. Let $T$ be the line passing through the points $P(-2,7)$ and $Q(2,-5)$. Let $F_1$ be the set of all pairs of circles $(S_1,S_2)$ such that $T$ is tangent to $S_1$ at $P$ and tangent to $S_2$ at $Q$, and also such that $S_1$ and $S_2$ touch each other at a point, say, $M$. Let $E_1$ be the set representing the locus of $M$ as the pair $(S_1,S_2)$ varies in $F_1$. Let the set of all straight line segments joining a pair of distinct points of $E_1$ and passing through the point $R(1,1)$ be $F_2$. Let $E_2$ be the set of the mid-points of the line segments in the set $F_2$. Then, which of the following statement(s) is (are) TRUE?

1. The point $(-2,7)$ lies in $E_1$.
2. The point $(\frac{4}{5},\frac{7}{5})$ does $\text{NOT}$ lie in $E_2$.
3. The point $(\frac{1}{2},1)$ lies in $E_2$.
4. The point $(0,\frac{3}{2})$ does $\text{NOT}$ lie in $E_1$.

Q. For $a>0$, let the curves $C_1:y^2=ax$ and $C_2:x^2=ay$ intersect at origin $O$ and a point $P$. Let the line $x=b(0<b<a)$ intersects the chord $OP$ and the $x$-axis at points $Q$ and $R$, respectively. If the line $x=b$ bisects the area bounded by the curves, $C_1$ and $C_2$, and the area of $\Delta OQR=\frac{1}{2}$, then '$a$' satisfies the equation:

1. $x^6-12x^3+4=0$
2. $x^6-12x^3-4=0$
3. $x^6+6x^3-4=0$
4. $x^6-6x^3+4=0$

Q. The equation of the image of the circle $x^2+y^2-6x-4y=0$ in the bisector of $2^{\text{nd}}$ and $4^{\text{th}}$ quadrant is

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

Q. Let $C:x^2+y^2-x-y-6=0$ and point $P(a-1,a+1)$ lies inside the circle $C$. If the line $x+y-2=0$ divides the circle in two segments, then

1. $P$ to lie in the larger segment of the intersection if $a\in (-1,1)$
2. $P$ to lie in the larger segment of the intersection if $a\in (1,2)$
3. $P$ to lie in the smaller segment of the intersection if $a\in (1,2)$
4. $P$ to lie in the smaller segment of the intersection if $a\in (-1,1)$

Q. The greatest distance of the point $P(9,7),$ from the circle $x^2+y^2-2x-2y-23=0,$ is units

Q. The orthocentre of the triangle formed by the lines $x+y=1,2x+3y=6$ and $4x-y+4=0$ lies in

1. first quadrant
2. second quadrant
3. third quadrant
4. fourth quadrant

Q. A point equidistant from the line 4x + 3y +10 = 0, 5x - 12y + 26 = 0 and 7x + 24y - 50 = 0 is ?

Q. For the circle $x^2+y^2+6x+8y=0,$ and the points $P(-3,-6)$ and $Q(4,-2),$

1. $Q$ lies inside and $P$ lies outside the circle
2. $P$ lies inside and $Q$ lies outside the circle
3. $P,Q$ lie outside the circle
4. $P,Q$ lie inside the circle

Q. $\overrightarrow{p}=w\hat{i}+x\hat{j}$ and $\overrightarrow{q}=y\hat{i}+z\hat{j}$ are two vectors in the first quadrant such that $|\overrightarrow{p}|=2|\overrightarrow{q}|=2r,r>0$ and $\overrightarrow{p}\cdot \overrightarrow{q}=0$. If $\overrightarrow{a}=w\hat{i}+2y\hat{j}$ and $\overrightarrow{b}=\frac{x}{2}\hat{i}+z\hat{j}$, then

1. $|\overrightarrow{a}|=r$
2. $|\overrightarrow{b}|=r$
3. $\overrightarrow{a}\cdot \overrightarrow{b}=0$
4. $|\overrightarrow{a}\times \overrightarrow{b}|=2r$

Q. From the point $P(2,1),$ a line of slope $m\in R$ is drawn so as to cut the circle $x^2+y^2=1$ in points $A$ and $B$. If the slope $m$ is varied, then the greatest possible value of $PA+PB$ is

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

Q. How many words can be formed from the letters of the word $\text{TRIANGLE}$? In how many of these does the word start with $T$ and end with $E$ ?

Q. If $d_1$ and $d_2$ are the longest and the shortest distances of the point $P(-7,2)$ from the circle $x^2+y^2-10x-14y-51=0,$ then the value of $d_1^2+d_2^2$ is