Equation of a Chord with a Given Middle Point

Q. The equation of the diameter of the circle $x^2+y^2+2x-4y-4=0$ which is parallel to $3x+5y-4=0$ is

1. $3x+5y=-7$
2. $3x+5y=7$
3. $3x+5y=9$
4. $3x+5y=1$

Q. The locus of midpoint of the chord of contact of $x^2+y^2=2$ from the points on $3x+4y=10$ is a circle whose centre is

1. $(\frac{4}{5},\frac{3}{5})$
2. $(\frac{3}{5},\frac{4}{5})$
3. $(\frac{4}{10},\frac{3}{10})$
4. $(\frac{3}{10},\frac{4}{10})$

Q. The equation to the locus of the midpoints of chords of the circle $x^2+y^2=r^2$ having a constant length 2l is

1. $x^2+y^2=l^2$
2. $x^2+y^2=r^2-l^2$
3. $x^2+y^2=r^2+l^2$
4. $x^2+y^2=4l^2$

Q.

The locus of the mid points of the chords of the circle $x^2+y^2-ax-by=0$ which subtend a right angle at $(\frac{a}{2},\frac{b}{2})$ is :

1. $ax+by=0$

2. $ax+by=a^2+b2$

3. $x^2+y^2-ax-by+\frac{a^2+b^2}{8}=0$

4. $x^2+y^2-ax-by+\frac{a^2+b^2}{8}=0$

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. If tangents $PA$ and $PB$ are drawn to $x^2+y^2=9$ from any arbitrary point $P$ on the line $x+y=25,$ then the locus of mid point of chord $AB$ is

1. $25(x^2+y^2)=9(x+y)$
2. $25(x^2+y^2)=3(x+y)$
3. $5(x^2+y^2)=3(x+y)$
4. $5(x^2+y^2)=9(x+y)$

Q. Let $x-2y+7=0$ be a chord of the circle $x^2+y^2-2x-10y+1=0$. If the midpoint of the chord is $P(\alpha ,\beta )$, then the value of $5|\alpha -\beta |$ is

Q. The equation of the chord of the circle $x^2+y^2=r^2$ passing through $(2,3)$ and farthest from the centre is

1. $x-3y+7=0$
2. $2x+y-7=0$
3. $2x+3y-13=0$
4. $x+y-5=0$

Q. Let $3x-y-3=0$ is a diameter of the circle $x^2+y^2-4x-6y+4=0$. If $L_1$ is the chord which is bisected by the given diameter line, then which of the following is/are true?

1. When $L_1$ is the largest possible chord of the circle, its possible equation is $x+3y-11=0$
2. When $L_1$ is the largest possible chord of the circle, its possible equation is $x-3y+8=0$
3. When $L_1$ is at a distance of $1$ from the centre, its equation is $x+3y-11+\sqrt{10}=0$
4. When $L_1$ is at a distance of $1$ from the centre, its equation is $x+3y-11-\sqrt{10}=0$