Pair of Tangents from a Point

Q. The equations of the tangents drawn from the point (0, 1) to the circle $x^2+y^2-2x+4y=0$ are

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

Q.

The length (in units) of the tangent from point $\left(5,1\right)$ to the circle $x^2+y^2+6x-4y-3=0$ is

1. $81$

2. $29$

3. $7$

4. $21$

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.

A circle is of the form $x^2+y^2+2gx+2fy+c=0$ and a pair of tangents are drawn from $(x_1,y_1)$ to the circle.The combined equation of tangents is $S{S}_1=T^2.$

Where $S=x^2+y^2+2gx+2fy+c$

$S_1=x_1^2+y_1^2+2g{x}_1+2f{y}_1+c$

$T=x{x}_1+y{y}_1+g(x+x_1)+f(y+y_1)+c$

1. True

2. False

Q. An infinite number of tangents can be drawn from $(1,2)$ to the circle $x^2+y^2-2x-4y+\lambda =0.$ Then $\lambda$ is

1. $-20$
2. Not possible
3. $5$
4. $20$

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. Let A be the centre of the circle $x^2+y^2$ - 2x - 4y - 20 = 0. Suppose that the tangents at the points B(1, 7) and D(4, - 2) on the circle meet at the point C. The area of the quadrilateral ABCD is

1. 75 sq. unit
2. 145 sq. unit
3. 150 sq. unit
4. 50 sq. unit

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

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. An infinite number of tangents can be drawn from $(1,2)$ to the circle $x^2+y^2-2x-4y+\lambda =0.$ Then

1. $\lambda =20$
2. $\lambda =-20$
3. $\lambda =5$
4. no such $\lambda$ exists

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. 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 range of values of 'a' such that the angle $\theta$ between the pair of tangents drawn from (a, 0) to the circle $x^2+y^2=1$ satisfies $\frac{\pi }{2}<\theta <\pi$, is

1. $(1,2)$

2. $(1,\sqrt{2})$
   

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

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

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