Equation of a Chord with a Given Middle Point

Q.

The locus of the midpoints of a chord of a circle $x^2+y^2=4$, which subtends a right angle at the origin, is

1. $x^2+y^2=1$

2. $x^2+y^2=2$

3. $x+y=1$

4. $x+y=2$

Q.

The equation of the circle which intersects circles $x^2+y^2+x+2y+3=0,x^2+y^2+2x+4y+5=0$ and

$x^2+y^2-7x-8y-9=0$ at right angle, will be

1. 

2. 

3. 

4. 

Q. Locus of mid point of chords of $x^2+y^2+2gx+2fy+c=0$ that pass through the origin, is

1. $x^2+y^2+2gx+2fy+c=0$
2. $x^2+y^2+gx+fy+c=0$
3. $x^2+y^2+gx+fy=0$
4. $2(x^2+y^2+gx+fy)+c=0$

Q.

Equation of the diameter of the circle $x^2+y^2-2x+4y=0$ which passes through the origin is

1. $x+2y=0$
   

2. $x-2y=0$

3. $2x+y=0$

4. $2x-y=0$

Q. If tangents are drawn from points on the hyperbola $\frac{x^2}{4}-\frac{y^2}{9}=1$ to the circle $x^2+y^2=4,$ then the locus of the mid-point of the chord of contact is

1. $x^2+y^2=\frac{x^2}{9}-\frac{y^2}{4}$
2. $(x^2+y^2{)}^2=\frac{x^2}{4}-\frac{y^2}{9}$
3. $(x^2+y^2{)}^2=16(\frac{x^2}{4}-\frac{y^2}{9})$
4. $(x^2+y^2{)}^2=9(\frac{x^2}{9}-\frac{y^2}{4})$

Q. The equation of diameter which bisects the chord $3x+y+5=0$ of the circle $x^2+y^2=16$ is

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

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. 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. Let circle be $2x(x-a)+y(2y-b)=0,a\ne 0,b\ne 0.$ Then the condition on $a$ and $b$ if two chords each bisected by the $x$-axis, can be drawn to the circle from $(a,\frac{b}{2})$, is

1. None of these
2. $a^2>2b^2$
3. $a^2<2b^2$
4. $a^2=2b^2$

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

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. 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 $x+3y-11-\sqrt{10}=0$

Q. $OPQR$ is a square and $M,N$ are the middle points of the sides $PQ$ and $QR$, respectively, then the ratio of the areas of the square and the $△OMN$ is

1. $4:1$
2. $8:3$
3. $2:1$
4. $7:3$

Q. A line $y=mx+1$ intersects the circle $(x-3{)}^2+(y+2{)}^2=25$ at the points $P$ and $Q$. If the midpoint of the line segment $PQ$ has $x$-coordinate $-\frac{3}{5}$, then which one of the following options is correct?

1. $2\le m<4$
2. $4\le m<6$
3. $6\le m<8$
4. $-3\le m<-1$

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. $x+y+3=0$

Q. Chords of contact are drawn from the points on a tangent of a hyperbola $x^2-y^2=16$ to a parabola $y^2=16x$. If all the chords of contact pass through a fixed point $Q$, then the locus of the point $Q$ for different tangents on hyperbola is an ellipse, whose length of latus rectum is .

Q. Locus of mid point of chords of $x^2+y^2+2gx+2fy+c=0$ that pass through the origin, is

1. $x^2+y^2+2gx+2fy+c=0$
2. $x^2+y^2+gx+fy+c=0$
3. $x^2+y^2+gx+fy=0$
4. $2(x^2+y^2+gx+fy)+c=0$

Q. Find the equation of the circle that passes through the points $\left(1.0\right),(-1,0)$ and $(0,1)$

1. $x^2-y^2=1$
2. $x^2+y^2=1$
3. None of these
4. $x^2+y^2=6$

Q. If $P(x,y,z)$ is a point on the line segment joining $Q(2,2,4)$ and $R(3,5,6)$ such that the projection of $\overrightarrow{OP}$ on the axes are $\frac{13}{5},\frac{19}{5},\frac{26}{5}$ respectively, then $P$ divides $QR$ in ratio

1. $2:3$
2. $1:3$
3. $3:2$
4. $3:1$

Q. If the area of the circle $4x^2+4y^2+8x-16y+\lambda =0$ is $9\pi$ sq. units, then the value of $\lambda$ is

1. $-4$
2. $4$
3. $16$
4. $-16$
5. $-8$

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 $x+3y-11-\sqrt{10}=0$

Q. The equation of diameter which bisects the chord $3x+y+5=0$ of the circle $x^2+y^2=16$ is

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

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 $x+3y-11-\sqrt{10}=0$

Q.

Chord of contact from any point on the line x + y = 4r to the circle $x^2+y^2=r^2$ (such that $r>0$) passes through the point (1, 1).What will be the value of r.

1. 3

2. 8

3. 5

4. 4

Q. The chord of the parabola $y=-a^2{x}^2+5ax-4$ touches the curve $y=\frac{1}{1-x}$ at the point $x=2$ and is bisected by that point. Find ${}^{\prime }{a}^{\prime }$.

Q. Find the scalar components and magnitude of the vector joining the points $P(x_1,y_1,z_1)$ and $Q(x_2,y_2,z_2)$.
.

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. 4

2. -1
   

3. 1
   

4. 2