Position of a Line with Respect to Circle

Q. The least distance of the line $8x-4y+73=0$ from the circle $16x^2+16y^2+48x-8y-43=0$ is

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

Q.

Let $O$ be the origin. Let $OP=xi+yj–k$ and $OQ=–i+2j+3xk,x,y\in R,x>0$, be such that $\left|PQ\right|=\sqrt{20}$ and the vector $OP$ is perpendicular to $OQ$. If $OR=3i+zk–7k,z\in R$, is coplanar with OP and OQ, then the value of $x^2+y^2+z^2$ is equal to:

1. $2$

2. $9$

3. $1$

4. $7$

Q.

If $3x+y+k=0$ is a tangent to the circle $x^2+y^2=10,$ then the values of $k$ are

1. $\pm 7$

2. $\pm 5$

3. $\pm 10$

4. $\pm 9$

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 which of the following is/are correct?

1. The cosine of the angle between the pair of tangent drawn from $A$ to the circle $S_1$ is $\frac{4}{5}$
2. The circumcentre of $△OAB$ is $(\frac{3}{2},3)$
3. The length of tangent from $A$ to the circle $S_2=0$ is $\frac{9}{2}\text{units}$
4. If one of the diameter of the circle $S_2=0$ is a chord to the circle $S_3=0$ whose centre is $(\frac{3}{2},3)$, then the radius of circle $S_3=0$ is $3$.

Q. A circle $C_1$ passes through the origin $O$ and has diameter $4$ on the positive $x$-axis. The line $y=2x$ gives a chord $OA$ of circle $C_1$ . Let $C_2$ be the circle with $OA$ as a diameter. If the tangent to $C_2$ at the point $A$ meets the $x$-axis at $P$ and $y$-axis at $Q$, then $QA:AP$ is equal to

1. $1:3$
2. $1:5$
3. $1:4$
4. $2:5$

Q. Let a circle $C$ touch the lines $L_1:4x-3y+K_1=0$ and $L_2:4x-3y+K_2=0,K_1,K_2\in R$. If a line passing through the centre of the circle $C$ intersects $L_1$ at $(-1,2)$ and $L_2$ at $(3,-6)$, then the equation of the circle $C$ is:

Q.

Radius of circle(x–5)(x–1) + (y – 7)(y – 4) = 0 is

1. 
2. 
3. 3
4. 4

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.

If the lines $a_{1}x+b_{1}y+c_1=0\text{and}{a}_{2}x+b_{2}y+c_2=0$ cut the coordinates axes in concyclic points, then

1. $a_1{b}_1=a_2{b}_2$
   

2. $\frac{a_1}{a_2}=\frac{b_1}{b_2}$
   

3. $a_1+a_2=b_1+b_2$
   

4. $a_1{a}_2=b_1{b}_2$

Q.

One of the diameter of the circle $x^2+y^2-12x+4y+6=0$ is given by

1. $x+y=0$

2. $x+3y=0$

3. $x=y$

4. $3x+2y=0$

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. 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 circumcenter of the triangle is $x+y-xy+k\sqrt{x^2+y^2}=0$, then the value of $k$ is

1. $2$
2. $1$
3. $3$
4. $-2$

Q. If a circle passing through the point $(-4,3)$ also touches the lines $x+y=2$ and $x-y=2$, then which of the following is/are true?

1. The centre of the circle is $(-10+3\sqrt{6},0)$
2. The centre of the circle is $(-10-3\sqrt{6},0)$
3. The radius of the circle is $r=6\sqrt{2}-3\sqrt{3}$ units
4. The radius of the circle is $r=6\sqrt{2}+3\sqrt{3}$ units

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. $5<k<25$
4. $9<k<25$

Q.

If the point $(\lambda ,\lambda +1)$ lies inside the region bounded by the curve $x=\sqrt{25-y^2}$ and y-axis, then $\lambda$ belongs to the interval

1. $(-1,3)$

2. $(-4,3)$
   

3. $(-\infty ,-4)\cup (3,\infty )$
   

4. none of these

Q. Equation of the circle, centred at $(1,–5)$ and touching the line $3x+4y=8,$ is

1. $x^2+y^2-2x+10y+1=0$
2. $x^2+y^2-4x+5y+25=0$
3. $x^2+y^2+2x-10y+26=0$
4. $x^2+y^2-10x+2y+1=0$

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. A circle touches the line $y=x$ at a point $P$ such that $OP=4\sqrt{2}$, where $O$ is the origin. The circle makes an intercept of $6\sqrt{2}$ units on line $x+y=0.$ Then the equation of the circle(s) is/are

1. $(x-9{)}^2+(y+1{)}^2-50=0$
2. $(x-1{)}^2+(y+9{)}^2-50=0$
3. $(x+1{)}^2+(y-9{)}^2-50=0$
4. $(x+9{)}^2+(y-1{)}^2-50=0$

Q. $C_1$ is a circle with centre at the origin and radius equal to $r$ and $C_2$ is a circle with centre at $(3r,0)$ and radius equal to $2r$. The number of common tangents that can be drawn to the two circles are

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

Q. The position of line $3x+4y-6=0$ with respect to the circle $x^2+y^2+2x+8y+1=0$ is

1. Tangent
2. lies outside the the circle
3. Diameter
4. Chord

Q.

If the line y = mx does not intersect the circle
$(x+10{)}^2+(y+10{)}^2=180$
then write the set of values taken by m

Q.

A tangent drawn to the curve $y=f\left(x\right)$ at $P(x,y)$ cuts the $x-axis$ and $y-axis$ at A and B respectively such that $BP:AP=3:1$given that $f\left(1\right)=1$, then

1. Equation of the curve is $x\left(\frac{dy}{dx}\right)-3y=0$

2. Normal at $(1,1)$ is $x+3y=4$

3. Curve passes through$\left(2,\frac{1}{8}\right)$

4. Equation of the curve is $x\left(\frac{dy}{dx}\right)+3y=0$

5. $3$ and $4$

Q.

The equation of the circle with center $(2,1)$ and touching the line $3x+4y=5$ is:

1. $x^2+y^2-4x-2y+5=0$

2. $x^2+y^2-4x-2y-5=0$

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

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

Q.

If the sum of squares of the intercept on the axes cut off by the tangent on the curve $x^{\frac{1}{3}}+y^{\frac{1}{3}}=a^{\frac{1}{3}},a>0$ at $(\frac{a}{8},\frac{a}{8})$ is $2,$ then value of $a$ is

Q. Assertion :If the perpendicular bisector of the line segment Joining $P(1,4)$ and $Q(K.3)$ has $y$-intercept $-4,$ then $k^2-16=0$ Reason: Locus of a point equidistant from two given points is the perpendicular bisector of the line joining the given points.

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. 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. 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.

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

1. 

2. 

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. $\text{(A)}\to \text{(P)},\text{(S)}$
2. $\text{(A)}\to \text{(P)},\text{(T)}$
3. $\text{(B)}\to \text{(P)},\text{(R)}$
4. $\text{(B)}\to \text{(Q)},\text{(T)}$

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. $5<k<25$