Pair of Tangents from a Point

Q. The equation of pair of tangents drawn to the circle $x^2+y^2-2x+4y+3=0$ from point $(6,-5)$ is

1. $7x^2+23y^2-30xy+66x-50y-73=0$
2. $7x^2+23y^2+30xy+66x+50y-73=0$
3. $7x^2+3y^2+30xy-66x+50y-73=0$
4. $3x^2+7y^2+30xy+66x+50y-73=0$

Q. Let the tangents at two points $A$ and $B$ on the circle $x^2+y^2-4x+3=0$ meet at origin $O(0,0)$. Then the area of the triangle $OAB$ is

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

Q. Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8x-6y-23=0.$ Let ${\Gamma }_A$ and ${\Gamma }_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles ${\Gamma }_A$ and ${\Gamma }_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and line passing through $A$ and $B,$ then the length of the line segment $AC$ is

Q. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

1. $20(x^2+y^2)+36x-45y=0$
2. $20(x^2+y^2)-36x+45y=0$
3. $36(x^2+y^2)-20y+45y=0$
4. $36(x^2+y^2)+20x-5y=0$

Q. There are three circles $C_1,C_2,C_3$ with radii $r_1,r_2,r_3(r_1<r_2<r_3)$ respectively touching each other externally as shown in the figure and $L_1,L_2$ are two common tangents to these circles from point $P$.


Which of the following is(are) correct?

1. $r_2=\frac{r_1^2+r_3^2}{r_1+r_3}$
2. $r_2^2=r_1{r}_3$
3. $DE=2\sqrt{r_1{r}_2}$
4. $DE=\sqrt{r_2^2-r_1^2}$

Q. Consider the circle $x^2+y^2-8x-18y+93=0$ with centre $C$ and point $P(2,5)$ outside it. From the point $P,$ a pair of tangents $PQ$ and $PR$ are drawn to the circle with $S$ as the midpoint of $QR$. The line joining $P$ to $C$ intersects the given circle at $A$ and $B$. Which of the following hold(s) good?

1. $CP$ is the arithmetic mean of $AP$ and $BP$.
2. $PR$ is the geometric mean of $PS$ and $PC$.
3. $PS$ is the harmonic mean of $PA$ and $PB$.
4. The angle between the two tangents from $P$ is ${\tan }^{-1}\left(\frac{3}{4}\right)$.

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

1. $x^2+y^2-6x-4y+19=0$
2. $x^2+y^2+4x-6y+19=0$
3. $x^2+y^2-2x+6y-20=0$
4. $x^2+y^2-4x-10y+19=0$

Q.

If O is the origin and OP, OQ are distinct tangents to the circle $x^2+y^2+2gx+2fy+c=0,$ the circumcentre of the triangle OPQ is

1. (-g, -f)
   

2. (g, f)
   

3. (-f, -g)
   

4. None of these

Q. $P$ is a variable point on the line $L=0$. Tangents are drawn to the circle $x^2+y^2=4$ from $P$ to touch it at $Q$ and $R$. The parallelogram $PQRS$ is completed.

If $L\equiv 2x+y-6=0$, then the locus of circumcentre of $△PQR$ is

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

Q. A pair of tangents is drawn to a unit circle with centre at the origin and these tangents intersect at $A$ enclosing an angle of ${60}^{\circ }.$ The area enclosed by these tangents and the arc of the circle is

1. $\frac{\pi }{3}-\frac{\sqrt{3}}{6}$ sq. units
2. $\sqrt{3}(1-\frac{\pi }{6})$ sq. units
3. $\frac{2}{\sqrt{3}}-\frac{\pi }{6}$ sq. units
4. $\sqrt{3}-\frac{\pi }{3}$ sq. units

Q. If the latus rectum of a hyperbola through one focus subtends ${60}^{\circ }$ angle at the other focus, then its eccentricity e is

1. 
2. 
3. 
4. 

Q.

Find the equations of the tangent and normal to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ at the point $(x_0,y_0)$.

Q. The locus of the point of intersection of two perpendicular tangents to the circle $x^2+y^2=a^2$ is

1. $x^2+y^2=\frac{a^2}{2}$
2. $x^2+y^2=\frac{a^2}{3}$
3. $x^2+y^2=2a^2$
4. $x^2+y^2=3a^2$

Q. Let $x^2+y^2-4x-2y-11=0$ be a circle. A pair of tangents from the point (4, 5) with a pair of radii form a quadrilateral of area ___.

Q. The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line $4x-5y=20$ to the circle $x^2+y^2=9$ is

1. $20(x^2+y^2)-36x+45y=0$
2. $20(x^2+y^2)+36x-45y=0$
3. $36(x^2+y^2)-20y+45y=0$
4. $36(x^2+y^2)+20x-5y=0$

Q.

The pair of tangents are drawn from an external point to a cicle. 2 radii are also drawn from the point of contact of those tangents to the centre. Then the quadrilateral formed by the tangents and the radii can always be inscribed in a circle.

1. True

2. False

Q. $P$ is a variable point on the line $L=0$. Tangents are drawn to the circle $x^2+y^2=4$ from $P$ to touch it at $Q$ and $R$. The parallelogram $PQRS$ is completed.

If $P\equiv (3,4)$, then coordinate of $S$ is

1. $(-\frac{46}{25},-\frac{63}{25})$
2. $(-\frac{51}{25},-\frac{68}{25})$
3. $(-\frac{46}{25},-\frac{68}{25})$
4. $(-\frac{68}{25},-\frac{51}{25})$

Q. Two lines $4x+2y=10$ and $2x-y=20$ are touching a circle whose radius is $\sqrt{5}$ units. Then the equation of the circle which is nearest to the $x$-axis, is

1. ${(x-6.25)}^2+{(y+2.5)}^2=5$
2. ${(x-6.25)}^2+{(y+7.5)}^2=5$
3. ${(x-8.75)}^2+{(y+7.5)}^2=5$
4. ${(x-3.75)}^2+{(y+7.5)}^2=5$

Q. Three circles, each of diameter $1$, are drawn each tangential to the others. A square enclosing the three circles is drawn so that two adjacent sides of the square are tangents to one of the circles and the square is as small as possible. The side length of this square is $a+\frac{\sqrt{b}+\sqrt{c}}{12}$ where $a,b,c$ are integers that are unique (except for swapping $b$ and $c.$) Find $a+b+c.$
(correct answer + 5, wrong answer 0)

Q. Find the point of Intersection of the circle $x^2+y^2=4$ with line passing through $A(1,0)$ and $B(3,4)$.

Q. Equation of pair of tangents drawn from $(4,3)$ to the circle $x^2+y^2=4$ is

1. $5x^2+12y^2+24xy-32x+24y-100=0$
2. $5x^2+12y^2-24xy+32x+24y-100=0$
3. $5x^2+12y^2-24xy-32x+24y+100=0$
4. $5x^2+12y^2+24xy+16x+24y-100=0$

Q. The locus of the point of intersection of the perpendicular tangents to the circles $x^2$ + $y^2$ = $a^2$, $x^2$ + $y^2$ = $b^2$ is

1. + =
2. + = -
3. + = +
4. + =

Q. The equation of the tangents to the circle $x^2+y^2+6x+6y+2=0$, which is parallel to $3x+4y+8=0$ are

1. $3x+4y-41=0,3x+4y+1=0$
2. $3x+4y+41=0,3x+4y+1=0$
3. $3x+4y+41=0,3x+4y-1=0$
4. $3x+4y-41=0,3x+4y-1=0$

Q.

Find the equation of pair of tangents drawn to the circle $x^2+y^2-4x+4y=0$ from the point (2, 2)

1. $4x^2+4y^2-16x+13y=0$

2. $x^2+y^2-17x+15y=0$

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

4. $3x^2+3y^2-17x-15y=0$

Q. Two lines $4x+2y=10$ and $2x-y=20$ are touching a circle whose radius is $\sqrt{5}$ units. Then the equation of the circle which is nearest to the $x$-axis, is

1. ${(x-6.25)}^2+{(y+2.5)}^2=5$
2. ${(x-6.25)}^2+{(y+7.5)}^2=5$
3. ${(x-8.75)}^2+{(y+7.5)}^2=5$
4. ${(x-3.75)}^2+{(y+7.5)}^2=5$

Q. $P$ is a variable point on the line $L=0$. Tangents are drawn to the circle $x^2+y^2=4$ from $P$ to touch it at $Q$ and $R$. The parallelogram $PQRS$ is completed.
If $P\equiv (6,8)$, then the area of $△QRS$ is

1. $\frac{196\sqrt{5}}{25}$ sq. unit
2. $\frac{196\sqrt{6}}{52}$ sq. unit
3. $\frac{192\sqrt{6}}{25}$ sq. unit
4. $\frac{196\sqrt{5}}{25}$ sq. unit

Q. If O is the origin and OP, OQ are the tangents from the origin to the circle $x^2+y^2-6x+4y+8=0$, the circumcenter of the triangle OPQ is

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

Q. There are three circles $C_1,C_2,C_3$ with radii $r_1,r_2,r_3(r_1<r_2<r_3)$ respectively touching each other externally as shown in the figure and $L_1,L_2$ are two common tangents to these circles from point $P$.


Which of the following is(are) correct?

1. $r_2=\frac{r_1^2+r_3^2}{r_1+r_3}$
2. $r_2^2=r_1{r}_3$
3. $DE=2\sqrt{r_1{r}_2}$
4. $DE=\sqrt{r_2^2-r_1^2}$

Q. Let the point $B$ be the reflection of the point $A(2,3)$ with respect to the line $8x-6y-23=0.$ Let ${\Gamma }_A$ and ${\Gamma }_B$ be circles of radii $2$ and $1$ with centres $A$ and $B$ respectively. Let $T$ be a common tangent to the circles ${\Gamma }_A$ and ${\Gamma }_B$ such that both the circles are on the same side of $T$. If $C$ is the point of intersection of $T$ and line passing through $A$ and $B,$ then the length of the line segment $AC$ is

Q. Equation of pair of tangents drawn from $(4,3)$ to the circle $x^2+y^2=4$ is

1. $5x^2+12y^2+24xy-32x+24y-100=0$
2. $5x^2+12y^2-24xy+32x+24y-100=0$
3. $5x^2+12y^2-24xy-32x+24y+100=0$
4. $5x^2+12y^2+24xy+16x+24y-100=0$