Position of a Line with Respect to Circle

Q.

Find the value of c if the line y = 3x + c is a tangent to the circle $x^2+y^2=10$

1. 10

2. -10

3. 100

4. -100

Q. The straight line $2x-3y=1$ divides the circular region $x^2+y^2\le 6$ into two parts. If $S=\{(2,\frac{3}{4}),(\frac{5}{2},\frac{3}{4}),(\frac{1}{4},\frac{-1}{4}),(\frac{1}{8},\frac{1}{4})\},$ then the number of point(s) in $S,$ lying inside the smaller part is

Q. If the circle $x^2+y^2-6x-10y+k=0$ does not touch or intersect the coordinate axes, and the point $(1,4)$ is inside the circle, then the range of $k$ is

1. $25<k<29$
2. $9<k<29$
3. $9<k<25$
4. $5<k<25$

Q.

The circle passing through $(1,-2)$ and touching the X-axis at$(3,0)$, also passes through the point

1. $(-5,2)$

2. $(2,-5)$

3. $(5,-2)$

4. $\left(-2,5\right)$

Q.

What is the condition for the line y = mx + c to be a secant of the circle $x^2+y^2=a^2$

1. $a^2(1+m^2)<c^2$

2. $a^2(1+m^2)>c^2$

3. $a^2(1+m^2)=c^2$

4. $c<a^2(1+m^2)$

Q.

Find the value of k if x + y + 5 = 0 is a tangent to the circle $x^2+y^2+10x+2ky+10=0$

1. $\sqrt{15}$

2. $-\sqrt{30}$

3. $\sqrt{30}$

4. None of these

Q. The equation of the circle passing through points of intersection of the circle $x^2+y^2-2x-4y+4=0$ and the line $x+2y=4$ and touches the line $x+2y=0$, is

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

Q. If circle $x^2+y^2-6x-10y+c=0$ does not touch (or) intersect the coordinates axes and the point $(1,4)$ is inside the circle, then the range of $c$ is

1. $(25,29)$
2. $(6,29)$
3. $(6,25)$
4. $R-(6,25)$

Q. If the line $x\cos \alpha +y\sin \alpha =p$ intersects the circle $x^2+y^2=4$ at $A$ and $B$ and chord $AB$ makes an angle of ${30}^{\circ }$ at a point on the circumference of circle then $3p^2=$

Q. If circle $x^2+y^2-6x-10y+c=0$ does not touch (or) intersect the coordinates axes and the point $(1,4)$ is inside the circle, then the range of $c$ is

1. $(6,25)$
2. $(25,29)$
3. $R-(6,25)$
4. $(6,29)$

Q.

The value of
$\tan {130}^{\circ }.\tan {140}^{\circ }$ is equal to

1. $1$

2. $\frac{1}{\sqrt{3}}$

3. $-1$

4. $1+\sqrt{3}$

Q. If circle $x^2+y^2-6x-10y+c=0$ does not touch (or) intersect the coordinates axes and the point $(1,4)$ is inside the circle, then the range of $c$ is

1. $R-(6,25)$
2. $(6,25)$
3. $(25,29)$
4. $(6,29)$

Q.

The value of 2$(si{n}^6{735}^{\circ }+co{s}^6{735}^{\circ })$ - 3 $(si{n}^4{735}^{\circ }+co{s}^4{735}^{\circ })$ + 1 is __.

Q. The line $y=mx$ intersects the circle $x^2+y^2-4x-4y=0$ and $x^2+y^2+6x-8y=0$ at points $A$ & $B$ (points being other than the origin). The range of $m$ such that the origin divides $AB$ internally is

1. $m>\frac{4}{3}$ or $m<-2$
2. $m>-1$
3. $-2<m<\frac{4}{3}$
4. $-1<m<\frac{3}{4}$

Q. $\text{List I}$ has four entries and $\text{List II}$ has five entries. Each entry of $\text{List I}$ is to be matched with one or more than one entries of $\text{List II}.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{Centre(s) of the circle(s) having radius}5\text{and}& \text{(P)}& (4,6)\\ & \text{touching the line}3x+4y-11=0\text{at}(1,2)\text{is (are)}& & \\ \text{(B)}& \text{End points of one of the diameters of}& \text{(Q)}& (1,1)\\ & x^2+y^2-6x-8y+20=0\text{are}& & \\ \text{(C)}& \text{The line equidistant from both the lines}& \text{(R)}& (3,-3)\\ & 4x+2y+2=0\text{and}6x+3y-21=0,& & \\ & \text{passes through}& & \\ \text{(D)}& \text{If one of the sides of a square is}3x-4y-12=0& \text{(S)}& (-2,7)\\ & \text{and its centre is}(0,0),\text{then its diagonal}& & \\ & \text{passes through}& & \\ & & \text{(T)}& (-2,-2)\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(Q)},\text{(R)},\text{(S)}$
2. $\text{(C)}\to \text{(Q)},\text{(T)}$
3. $\text{(D)}\to \text{(P)},\text{(Q)}$
4. $\text{(D)}\to \text{(R)},\text{(T)}$