Direct Common Tangent

Q. the common tangent to the circles $x^2+y^2-6x=0$ and $x^2+y^2+2x=0$ forms

1. Right angle triangle
2. Right angle isosceles triangle
3. isosceles triangle
4. Equilateral triangle

Q.

Let the tangent to the circle $x^2+y^2=25$ at the point $R(3,4)$ meet the $x$-axis and $y$-axis at points $P\&Q$, respectively. If $r$ is the radius of the circle passing through the origin $O$ and having a centre at the incentre of the triangle $OPQ$, then$r^2$ is equal to:

1. $\frac{625}{72}$

2. $\frac{585}{66}$

3. $\frac{125}{72}$

4. $\frac{529}{64}$

Q. A circle $C$ touches the line $x=2y$ at the point $(2,1)$ and intersects the circle $C_1:x^2+y^2+2y-5=0$ at two points $P$ and $Q$ such that $PQ$ is a diameter of $C_1.$ Then the diameter of $C$ is

1. $\sqrt{285}$
2. $15$
3. $4\sqrt{15}$
4. $7\sqrt{5}$

Q. Consider a circle $C$ which touches the $y-$axis at $(0,6)$ and cuts off an intercept $6\sqrt{5}$ on the $x-$axis. Then the radius of the circle $C$ is equal to

1. $\sqrt{53}$
2. $9$
3. $8$
4. $\sqrt{82}$

Q. Let the circle $S:36x^2+36y^2-108x+120y+C=0$ be such that it neither intersects nor touches the co-ordinate axes. If the point of intersection of the lines, $x-2y=4$ and $2x-y=5$ lies inside the circle $S$, then:

1. $\frac{25}{9}<C<\frac{13}{3}$
2. $81<C<156$
3. $100<C<156$
4. $100<C<165$

Q. A circle with centre at $(15,-3)$ is tangent to $y=\frac{x^2}{3}$ at a point in the first quadrant. The radius of the circle is equal to $a\sqrt{5}$ where $a$ is

Q. The equation of common tangent to the circles $x^2+y^2=4$ and $x^2+y^2-6x-8y-24=0$ is

1. $3x-4y+10=0$
2. $3x-4y-10=0$
3. $3x+4y+10=0$
4. $3x+4y-10=0$

Q. The area of the region bounded by the curve y = y = tan x, tangent drawn to the curve at $x=\frac{\pi }{4}$ and the x-axis is

1. $log2-\frac{1}{4}$
2. $\frac{1}{2}log2-\frac{1}{4}$
3. $log2-\frac{1}{2}$
4. None

Q. If the circles $x^2+y^2-4x-6y-12=0$ and $5(x^2+y^2)-8x-14y-32=0$ touch each other then their point of contact is

1. $(0,0)$
2. $(1,1)$
3. $(-1,-1)$
4. $(0,-1)$

Q. If two circles of radius $25$ units and $16$ units touch each other externally, then the radius of the circle which touches both of them externally and also their direct common tangent is

1. $20\text{units}$
2. $\frac{400}{81}\text{units}$
3. $\frac{41}{2}\text{units}$
4. $\frac{9}{2}\text{units}$

Q. The point(s) on the ellipse $16x^2+11y^2=256$
where the common tangent to it and the circle $x^2+y^2-2x=15$ touch is

1. $(2,8\sqrt{\frac{3}{11}})$
2. $(2,8\sqrt{\frac{3}{11}})$
3. $(-2,-8\sqrt{\frac{3}{11}})$
4. $(2,-8\sqrt{\frac{3}{11}})$

Q. If two circles with centres at $(a,0)$ and $(-a,0)$ having radii $b$ and $c$ units respectively such that $a>b>c$. Then the point of contacts of common tangents to these two circles will always lie on

1. $x^2+y^2=a^2\pm bc$
2. $x^2-y^2=a^2\pm bc$
3. $x^2+2y^2=a^2\pm bc$
4. $2x^2+y^2=a^2\pm bc$

Q. A variable circle always touches the line $y=x$ at the origin. If all the common chords of the given circle and $x^2+y^2+6x+8y-7=0$ passes through a fixed point $(a,b)$, then $a+b$ is

Q.

Find the equations of direct common tangents for two circles

$x^2+y^2+6x-2y+1=0,$ $x^2+y^2-2x-6y+9=0$

1. $35x^2+12xy-18x=0$

2. $35y^2+12xy-18y=0$

3. $3x^2-4xy+16y-12x=0$

4. $3y^2-4xy+16x-12y=0$

Q. Let $S_1$ and $S_2$ be two unit circles with centres at $C_1(0,0)$ and $C_2(1,0)$ respectively. Let $S_3$ be another circle of unit radius, passing through $C_1$ and $C_2$ and its centre is above the $x$-axis. If equation of common tangent to $S_1$ and $S_3$, which does not pass through $S_2$, is $ax+by+2=0$, then the value of $a^2-b$ is

Q. If the length of the common chord of two circles of radii $3$ and $4$ units, which intersect orthogonally is $\frac{k}{5},$ then the value of $k$ is

1. $16$
2. $7$
3. $24$
4. $25$

Q. Let two circles touch each other externally and $P$ is the point of intersection of there direct common tangents. If the direct common tangents from $P$ touches the circles at $A$ and $B$ on same side and $PA=AB=4$, then the radius of smaller circle (in units) is

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

Q.

(a, b) is the mid point of the chord $\overline{AB}$ of the circle $x^2+y^2=r^2$. The tangent at A, B meet a C. then area of $\Delta ABC$

1. 

2. 

3. 

4. 

Q. Tangent drawn to circle $\mid z\mid =2$ at $A\left(z_1\right)$ and normal at $B\left(z_2\right)$ meet at the point $P\left(z_p\right)$, then

1. none of these
2. 
   $zp=\frac{2\mid z_{1}mi{d}^2{z}_2}{z_1{\overline{z}}_2+\overline{z_1}{z}_2}$
3. 
   $zp=\frac{2\mid z_1{\mid }^2{z}_2}{z_1{\overline{z}}_2+\overline{z_1}{z}_2}$
4. $zp=\frac{2\mid z_1{\mid }^2{z}_1}{z_1{\overline{z}}_2+\overline{z_1}{z}_2}$

Q. If the circles $x^2+y^2-4x-6y-12=0$ and $5(x^2+y^2)-8x-14y-32=0$ touch each other then their point of contact is

1. $(0,0)$
2. $(1,1)$
3. $(-1,-1)$
4. $(0,-1)$

Q. Area of the hexagon formed by points of contact of direct common tangents to the circle $(x-1{)}^2+(y-3{)}^2=9$, $(x-7{)}^2+(y+5{)}^2=25$ with the two centres is

1. $30\sqrt{6}\text{sq. units}$
2. $45\sqrt{6}\text{sq. units}$
3. $25\sqrt{6}\text{sq. units}$
4. $32\sqrt{6}\text{sq. units}$

Q. If $y=mx-b\sqrt{1+m^2}$ is a common tangent to $x^2+y^2=b^2$ and $(x-a{)}^2+y^2=b^2$, where $a>2b>0$, then the positive value of $m$ is

1. $\frac{2b}{\sqrt{a^2-4b^2}}$
2. $\frac{\sqrt{a^2-4b^2}}{2a}$
3. $\frac{2b}{a-2b}$
4. $\frac{b}{a-2b}$

Q. Let two circles touch each other externally and $P$ is the point of intersection of there direct common tangents. If the direct common tangents from $P$ touches the circles at $A$ and $B$ on same side and $PA=AB=4$, then the radius of smaller circle (in units) is

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

Q. Tangents drawn from the point P(1, 8) to the circle $x^2+y^2-6x-4y–11=0$ touch the circle at the points A and B. The equation of the circum circle of the trianglePAB is

1. 
2. 
3. 
4. 

Q. The common tangents to the circles $x^2+y^2-6x=0$ and $x^2+y^2+2x=0$ is/are

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

Q. A possible equation of $L$ is

1. $x+\sqrt{3}y=5$
2. $x-\sqrt{3}y=1$
3. $x+\sqrt{3}y=1$
4. $x-\sqrt{3}y=-1$

Q. Three circles of radii $a,b,c(a<b<c)$ touch each other externally. If they have $x$-axis as a common tangent, then:

1. $a,b,c$ are in A.P.
2. $\sqrt{a},\sqrt{b},\sqrt{c}$ are in A.P.
3. $\frac{1}{\sqrt{b}}=\frac{1}{\sqrt{a}}+\frac{1}{\sqrt{c}}$
4. $\frac{1}{\sqrt{a}}=\frac{1}{\sqrt{b}}+\frac{1}{\sqrt{c}}$

Q. If $y=mx-b\sqrt{1+m^2}$ is a common tangent to $x^2+y^2=b^2$ and $(x-a{)}^2+y^2=b^2$, where $a>2b>0$, then the positive value of $m$ is

1. $\frac{b}{a-2b}$
2. $\frac{\sqrt{a^2-4b^2}}{2a}$
3. $\frac{2b}{\sqrt{a^2-4b^2}}$
4. $\frac{2b}{a-2b}$

Q. Let $P$ and $Q$ be points on the circle $x=a\cos \theta ,y=a\sin \theta$, such that the value of their parameters $\left(\theta \right)$ differs by $\frac{\pi }{6}$. The locus of the point of intersection of tangents to the circle at $P$ and $Q$ is

1. $x^2+y^2=(8-4\sqrt{3})a^2$
2. $x^2+y^2=(8-4\sqrt{2})a^2$
3. $x^2+y^2=(6-4\sqrt{3})a^2$
4. $x^2+y^2=(6-4\sqrt{2})a^2$

Q. The circle $x^2+y^2-6x-6y+9=0$ is inscribed in a triangle which has two of its sides along the coordinate axes. The locus of the circumcentre of the triangle is $x+y-\frac{b}{a+b}xy+a(x^2+y^2{)}^{\frac{1}{b}}=0$. Find $a+b$