Pair of Tangents from a point

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

1. 
2. 
3. 
4. 

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. 

2. 

3. 

4. 

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. From a point $P$ tangents drawn to the circles $x^2+y^2+x-3=0,3x^2+3y^2-5x+3y=0$ and $4x^2+4y^2+8x+7y+9=0$ are of equal length. Find the equation of the circle through $P$ which touches the line $x+y=5$ at the point $(6,-1)$.

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

Q. The angle between a pair of tangents drawn from a point P to the circle $x^2+y^2+4x-6y+9si{n}^2\alpha +13co{s}^2\alpha =0is2\alpha$. The equation of the locus of the point P is

1. 
2. 
3. 
4. 

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. State True or False:
On subtracting $cab-4cad-cbd$ from $3abc+5bcd-cda$, the answer is $2abc+3cad+6bcd$.

1. True
2. False

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

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

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

Q. The equations of the tangents drawn from the origin to the circle $x^2+y^2-2rx-2hy+h^2=0$ are

1. x = 0, y = 0
2. y = 0, x = 4
3. $(h^2-r^2)x-2rhy=0,x=0$
4. $(h^2-r^2)x+2rhy=0,x=0$

Q. Find the equation of circles passing through the point $(2,8)$, touching the lines $4x-3y-24=0$ and $4x+3y-42=0$ and having x-co-ordinate of the centre of the circle less than or equal to $8$.