Length of Chord of Contact

Q. Let tangents are drawn from $P(3,4)$ to the circle $x^2+y^2=a^2,$ touching the circle at $A$ and $B.$ If area of $△PAB$ is $\frac{192}{25}$ sq. units, then the absolute value of $a$ is

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(I)}& \text{The length of the common chord of}& & \\ & \text{two circle of radii}3\text{and}4\text{units which}& & \\ & \text{intersect orthogonally is}\frac{k}{5},\text{then k is}& \text{(P)}& 1\\ \text{(II)}& \text{The circumference of the circle}& & \\ & x^2+y^2+4x+12y+p=0\text{is bisected}& & \\ & \text{by the circle}{x}^2+y^2-2x+8y-q=0,& \text{(Q)}& 2\\ & \text{then}p+q\text{equals to}& & \\ \text{(III)}& \text{The number of distinct chords of}& & \\ & x^2+y^2+2x+4y=0\text{which passes}& \left(R\right)& 12\\ & \text{through the point}(2,1)\text{and bisected}& & \\ & \text{by}x\text{- axis is}& & \\ \text{(IV)}& \text{One of the diameter of the circle}& \left(S\right)& 24\\ & \text{circumscribing the rectangle}ABCD& & \\ & \text{is}4y=x+7.\text{If}A\text{and}B\text{are}(-3,4)& & \\ & \text{and}(5,4)\text{respectively, then the area}& & \\ & \text{of the rectangle}ABCD\text{is}& & \\ & & \left(T\right)& 32\\ & & \left(U\right)& 36\end{array}$

Which of the following option is INCORRECT ?

1. $\left(I\right)\to \left(R\right)$
2. $\left(II\right)\to \left(U\right)$
3. $\left(III\right)\to \left(P\right)$
4. $\left(IV\right)\to \left(T\right)$

Q. The length of the chord $AB$ of the circle $x^2+y^2-6x+8y-13=0$ whose midpoint is $(2,-3)$ is (units)

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

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

Q. $\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(I)}& \text{The length of the common chord of}& & \\ & \text{two circle of radii}3\text{and}4\text{units which}& & \\ & \text{intersect orthogonally is}\frac{k}{5},\text{then k is}& \text{(P)}& 1\\ \text{(II)}& \text{The circumference of the circle}& & \\ & x^2+y^2+4x+12y+p=0\text{is bisected}& & \\ & \text{by the circle}{x}^2+y^2-2x+8y-q=0,& \text{(Q)}& 2\\ & \text{then}p+q\text{equals to}& & \\ \text{(III)}& \text{The number of distinct chords of}& & \\ & x^2+y^2+2x+4y=0\text{which passes}& \left(R\right)& 12\\ & \text{through the point}(2,1)\text{and bisected}& & \\ & \text{by}x\text{- axis is}& & \\ \text{(IV)}& \text{One of the diameter of the circle}& \left(S\right)& 24\\ & \text{circumscribing the rectangle}ABCD& & \\ & \text{is}4y=x+7.\text{If}A\text{and}B\text{are}(-3,4)& & \\ & \text{and}(5,4)\text{respectively, then the area}& & \\ & \text{of the rectangle}ABCD\text{is}& & \\ & & \left(T\right)& 32\\ & & \left(U\right)& 36\end{array}$

Which of the following option is CORRECT ?

1. $\left(I\right)\to \left(R\right)$
2. $\left(II\right)\to \left(S\right)$
3. $\left(III\right)\to \left(Q\right)$
4. $\left(IV\right)\to \left(T\right)$

Q. Let the tangents drawn from the origin to the circle, $x^2+y^2-8x-4y+16=0$ touch it at the points $A$ and $B$. The $(AB{)}^2$ is equal to :

1. $\frac{32}{5}$
2. $\frac{64}{5}$
3. $\frac{52}{5}$
4. $\frac{56}{5}$