Equation of a Chord Joining Two Points with Circle in Parametric Form

Q. The straight line x-2y+1=0 intersects the circle $x^2+y^2=25$ at points A and B, then the coordinates of points of intersection of tangents drawn at A and B are

1. (-25, 50)
2. (25, -50)
3. (-5, 25)
4. (5, -25)

Q. Circles $C_1$ & $C_2$ externally touch each other and they both internally touch another circle $C_3$. The radii of $C_1$ & $C_2$ are $4$ and $10$ respectiveley, and the centers of the three circles are collinear. A chord of $C_3$ is a transverse common tangent to the circles $C_1$ and $C_2$. Find the length of the chord

1. $4\sqrt{10}$
2. $14$
3. $8\sqrt{5}$
4. None of these

Q. Match the following by approximately matching the lists based on the information given in Column I and Column II

$\begin{array}{cc}Column1& Column2\\ \text{a. The length of the common chord of two circles of radii}3\text{and}& \text{p.}1\\ 4\text{units which intersect orthogonally is}\frac{k}{5},\text{then}k\text{is equal to}& \\ \text{b. The circumference of the circle}{x}^2+y^2+4x+12y+p=0& \text{q.}24\\ \text{is bisected by the circle}{x}^2+y^2-2x+8y-q=0\text{, then}& \\ p+q\text{is equal to}& \\ \text{c. Number of distinct chords of the circle}2x(x-\sqrt{2})+y(2y-1)& \text{r.}32\\ =0\text{chords are passing through the point}(\sqrt{2},\frac{1}{2})\text{and are}& \\ \text{bisected on x-axis is}& \\ \text{d. One of the diameters of the circle circumscribing the rectangle}& \text{s.}36\\ ABCD\text{is}4y=x+7\text{. If}A\text{and}B\text{are the points}(-3,4)\text{and}& \\ (5,4)\text{respectively,}\text{then the area of rectangle is}& \end{array}$

1. $a-q,b-s,c-p,d-r$
2. $a-p,b-s,c-q,d-r$
3. $a-q,b-r,c-p,d-s$
4. $a-r,b-s,c-p,d-q$

Q. The locus of midpoint of chord of the circle $x^2+y^2-2x-2y-2=0$, which makes an angle of ${120}^{\circ }$ at the centre, is

1. $x^2+y^2-x-y+3=0$
2. $x^2+y^2-2x-2y+4=0$
3. $x^2+y^2-4x-4y+8=0$
4. $x^2+y^2-2x-2y+1=0$

Q. Three concentric circles of which biggest is $x^2+y^2=1$, have their radii in A.P. If the line $y=x+1$ cuts all the three circles in real and distinct points, then the interval in which the common difference of $AP$ will lie, is

1. $(0,\frac{\sqrt{2}-1}{2\sqrt{2}})$
2. $(0,\frac{\sqrt{2}+1}{2\sqrt{2}})$
3. $(0,\frac{\sqrt{2}-1}{\sqrt{2}})$
4. $(0,\frac{\sqrt{2}-1}{2})$

Q.

What is the equation of the chord centered at (1, 2) in the circle $x^2+y^2-4x-6y-10=0$

1. $x-y-3=0$

2. $x-y+3=0$

3. $x+y-3=0$

4. $x+y+3=0$