Length of Common Chord

Q. Two circles with equal radii are intersecting at the points $(0,1)$ and $(0,-1)$. The tangent at the point $(0,1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

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

Q. If the circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $5$ in such a manner that common chord is of maximum length and has a slope equal to $\frac{3}{4}$, then the absolute sum of coordinates of the centre $C_2$ is

Q. If the angle of intersection at a point where the two circles with radii $5cm$ and $12cm$ intersect is $90°,$ then the length $\left(\text{in}cm\right)$ of their common chord is :

1. $\frac{13}{5}$
2. $\frac{120}{13}$
3. $\frac{13}{2}$
4. $\frac{60}{13}$

Q. If the circles $x^2+y^2+5Kx+2y+K=0$ and $2(x^2+y^2)+2Kx+3y-1=0,(K\in R),$ intersect at the points $P$ and $Q$, then the line $4x+5y-K=0$ passes through $P$ and $Q$, for :

1. exactly one value of $K$
2. exactly two values of $K$
3. infinitely many values of $K$
4. no value of $K$

Q. The length of a common internal tangent to two circles is $7$ and a common external tangent is $11$. If the product of the radii of the two circles is $p$, then the value of $\frac{p}{2}$ is

Q. A rhombus is inscribed in the region common to the two circles $x^2+y^2-4x-12=0$ and $x^2+y^2+4x-12=0$ with two of its vertices on the line joining the centers of the circles. The area of the rhombus is

1. $6\sqrt{3}$ sq. units
2. $8\sqrt{3}$ sq. units
3. $4\sqrt{3}$ sq. units
4. $2\sqrt{3}$ sq. units

Q. Let $P(a,b)$ be a point in the first quadrant. Circles are drawn through $P$ touching the coordinate axes such that the length of common chord for the smaller circle is maximum. If possible values of $\frac{a}{b}$ is $k_1$ and $k_2,$ then $k_1+k_2$ is equal to

Q. For the given circles $x^2+y^2+2x+4y-20=0$ and $x^2+y^2+6x-8y+10=0$, which of the following is/are correct?

1. The number of common tangents is $2.$
2. The number of common tangents is $3.$
3. Length of the common tangent is $(1500{)}^{1/4}$ units
4. Length of the common tangent is $(1000{)}^{1/4}$ units

Q. Let any circle $S$ passes through the point of intersection of lines $\sqrt{3}(y-1)=x-1$ and $y-1=\sqrt{3}(x-1)$ and having its centre on the acute angle bisector of the given lines. If the common chord of $S$ and the circle $x^2+y^2+4x-6y+5=0$ passes through a fixed point, then the fixed point is

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

Q. Let $P(1,1)$ be a point inside the circle $x^2+y^2+2x+2y-8=0.$ The chord $AB$ is drawn passing through the point $P.$ If $\frac{PA}{PB}=\frac{\sqrt{5}-2}{\sqrt{5}+2},$ then equation of chord $AB$ is

1. $y=2x+1$
2. $y=x$
3. $y=-x$
4. $y=\sqrt{2}x+1$

Q. If the length of the chord of the circle, $x^2+y^2=r^2(r>0)$ along the line, $y-2x=3$ is $r,$ then $r^2$ is equal to

1. $12$
2. $\frac{24}{5}$
3. $\frac{9}{5}$
4. $\frac{12}{5}$

Q. For the given circles $x^2+y^2+2x+4y-20=0$ and $x^2+y^2+6x-8y+10=0$, which of the following is/are correct?

1. The number of common tangents is $2.$
2. The number of common tangents is $3.$
3. Length of the common tangent is $(1500{)}^{1/4}$ units
4. Length of the common tangent is $(1000{)}^{1/4}$ units

Q. The line $x+y=0$ bisects two chords drawn from the point $(\frac{1+k\sqrt{2}}{2},\frac{1-k\sqrt{2}}{2})$ to the circle $x^2+y^2-\left(\frac{1+k\sqrt{2}}{2}\right)x-\left(\frac{(1-k\sqrt{2})}{2}\right)y=0.$ Then the minimum integral value of $|k|$ is

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. Consider $f:R\to R,f\left(x\right)=|x^2-5|x|+6|$. Then which of the following is NOT true?

1. f(x) is non differentiable at 5 points
2. f(x) has local minima at x = 0
3. f(x) = R has 4 solutions for $R\in (\frac{1}{4},6)\cup$ {0}
4. f(x) = R has 8 solutions for $R\in (0,\frac{1}{3})$

Q. $P(a,b)$ is a point in the first quadrant. Circles are drawn through $P$ touching the coordinate axes such that the length of common chord of these circles is maximum, if possible values of $a/b$ is $k_1$ and $k_2$, then $k_1+k_2$ is equal to

Q. Two circles with equal radii are intersecting at the points $(0,1)$ and $(0,-1)$. The tangent at the point $(0,1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

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

Q. If $5,7,6$ are the sums of the $x,y$ intercepts; $y,z$ intercepts, $z,x$ intercepts respectively of a plane then the perpendicular distance from the origin to that plane is

1. $\frac{144}{61}$
2. $\frac{12}{\sqrt{61}}$
3. $\frac{61}{144}$
4. $\frac{\sqrt{61}}{12}$

Q. Two circles with equal radii are intersecting at the points $(0,1)$ and $(0,-1)$. The tangent at the point $(0,1)$ to one of the circles passes through the centre of the other circle. Then the distance between the centres of these circles is:

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

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

Q. The area occupied between the curve $xy=a^2,$ the vertical lines $x=a,x=4a(a>0)$ is

1. $a^2\ln 2$
2. $2a^2\ln 2$
3. $a\ln 2$
4. $2a\ln 2$

Q. Two rods of lengths $a$ and $b$ slide along the $x$- and the $y$-axis, respectively, in such a manner that their ends are concyclic. Find the locus of the center of the circle passing through the endpoints.

Q. If the circle $x^2+y^2=a^2$ intersects the hyperbola $xy=c^2$ in four points $P(x_1,y_1),Q(x_2,y_2),R(x_3,y_3),S(x_4,y_4)$, then which of the following need not hold

1. 
   
2. 
   
3. 
   
4. 
   

Q. Circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $6$ in such a manner that their common chord is of maximum length and has slope equal to $\frac{1}{2}.$ Then the co-ordinates of the centre of the circle(s) $C_2$ is (are)

1. $(-2,-4)$
2. $(2,4)$
3. $(-2,4)$
4. $(2,-4)$

Q. The locus of the pole of the chords of the standard circle which subtended a right angle at (h, k) is

Q.

If area of the triangle formed by the lines $y^2-9xy+18x^2=0$ and y=9 is A sq. unit find the value of 4A.

__

Q. Circle $C_1:x^2+y^2=16$ intersects another circle $C_2$ of radius $6$ in such a manner that their common chord is of maximum length and has slope equal to $\frac{1}{2}.$ Then the co-ordinates of the centre of the circle(s) $C_2$ is (are)

1. $(-2,-4)$
2. $(2,4)$
3. $(-2,4)$
4. $(2,-4)$

Q. A man is know to speak the truth $3$ out of $4$ times. He throws a die and reports that it is a six. the probability that it is actually a six is

1. $\frac{3}{8}$
2. $\frac{1}{5}$
3. $\frac{3}{4}$
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. Area enclosed between the curve $y=1-x^2$

1. $\frac{4}{3}$ sq. units
2. $\frac{2\sqrt{2}}{3}$ sq. units
3. $\frac{3\sqrt{2}}{3}$ sq units
4. None of these