Length of Common Chord

Q.

The distance between two parallel chords each of length 10 units is 24 units. Then the Radius of the Circle is

1. 5

2. 12

3. 13

4. 30

Q.

Find the equation of the hyperbola, the length of whose latustrectum is 8 and eccentricity is $3/\sqrt{5}$.Also determine the equation of directrices.

Or

Find the equation of the ellipse whose axes are along the coordinate axes, vertices are $\pm 5,0)$and foci at $(\pm 4,0)$. Also determine the length of major and minor axes.

Q.

The circle $x^2+y^2-6x-4y+9=0$ bisects the circumference of the circle $x^2+y^2-(\lambda +4)x-(\lambda +2)y+(5\lambda +3)=0if\lambda$ is equal to

1. -1
   

2. 1
   

3. 2
   

4. 4

Q. A fair die is rolled. The probability that the first time 1 occurs at an even throw, is

1. $\frac{1}{6}$
2. $\frac{5}{11}$
3. $\frac{6}{11}$
4. $\frac{5}{36}$

Q. $PQRS$ is a quadrilateral in which the diagonals intersect each other at O. Prove that $PQ+QR+RS+SP<2(PR+QS)$

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. 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. $8\sqrt{3}$ sq. units
2. $4\sqrt{3}$ sq. units
3. $6\sqrt{3}$ sq. units
4. $2\sqrt{3}$ sq. 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 PQR be an isosceles right angled triangle, right angled at P (2, 1). If the equation of the line QR is $2x+y=3$, then the equation representing the pair of lines PQ and PR is:

1. $3x^2-3y^2+8xy+20x+10y+25=0$
2. $3x^2-3y^2+8xy-20x-10y+25=0$
3. $3x^2-3y^2+8xy+10x+15y+20=0$
4. $3x^2-3y^2-8xy-10x-15y-20=0$

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 $K$