Equation of Normal for General Equation of a Circle

Q. The line $2x-y+1=0$ is tangent to a circle at point $(2,5)$ and the centre of the circle lies on $x-2y=4.$ Then the radius of the circle is ____ units.

1. $3\sqrt{5}$
2. $5\sqrt{3}$
3. $2\sqrt{5}$
4. $5\sqrt{2}$

Q. The equation of the normals to the circle $x^2+y^2-8x-2y+12=0$ at the points whose ordinate is $-1$ is/are

1. $2x+y-9=0$
2. $2x-y-7=0$
3. $2x+y-7=0$
4. $2x-y-9=0$

Q. If $N$ is the normal to the circle $x^2+y^2+4x+6y-39=0$ at the point $(2,3)$, then it meets the circle again at

1. $(-6,3)$
2. $(2,-9)$
3. $(2,9)$
4. $(-6,-9)$

Q. The line 4x-3y=-12 is the tangent at point A(-3, 0) and the line 3x+4y=16 is the tangent at the point B(4, 1) to a circle. The equation of circle is .

1. $(x+1{)}^2+(y+3{)}^2=25$
2. $(x+1{)}^2+(y-3{)}^2=25$
3. $(x-1{)}^2+(y+3{)}^2=25$
4. $(x-1{)}^2+(y-3{)}^2=25$

Q. The equation of the normal to the circle $x^2+y^2-2x=0$ parallel to the line $x+2y=3$ is

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

Q. If the line $ax+by=2$ is a normal to the circle $x^2+y^2-4x-4y=0$ and a tangent to the circle $x^2+y^2=1,$ then

1. $a=\frac{1}{2},b=\frac{1}{2}$
2. $a=\frac{1-\sqrt{7}}{2},b=\frac{1+\sqrt{7}}{2}$
3. $a=\frac{1}{4},b=\frac{3}{4}$
4. $a=1,b=2$

Q.

Find the equation of the normal to the circle $x^2+y^2-6x-8y-24=0$ at the point (4, 6).

1. 2x -y = 2

2. 3x + y = 18

3. 3x - y = 12

4. -3y + 12 = 0

Q. The equation of the normal to the circle $x^2+y^2-2x=0$ parallel to the line x + 2y = 3 is .

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

Q. From a point $P$ on the normal $y=x+c$ of the circle $x^2+y^2-2x-4y+5-k^2=0,$ two tangents are drawn to the same circle touching it at point $B$ and $C.$ If the area of the quadrilateral $OBPC$ (where $O$ is the centre of the cicrle ) is $36$ sq. units and ponit $P$ is at a distance of $|k|(\sqrt{2}-1)$ from the circle, then the possible value(s) of $k$ is/are

1. $6$
2. $6\sqrt{2}$
3. $-6$
4. $4\sqrt{2}$

Q.

If the equation of normal to the circle $x^2+y^2-16x-12y+99=0$, which is also a tangent to $x^2+y^2=4$ is ax + by + c = 0, find the value of $\left[|\right(\frac{a}{b}\left)|\right]$, where [x] is the greatest integer function

1. 0

2. 1

3. 2

4. 4

Q. The circle $C$ passing through the origin, has the line $3x+4y=0$ as its tangent at the origin. If the image of the centre of $C$ w.r.t the line $\frac{x}{\frac{25}{3}}+\frac{y}{\frac{25}{4}}=1$ lies on the line $3x+4y=0,$ then the equation of $C$ is

1. $(x-6{)}^2+(y-8{)}^2=100$
2. $(x-6{)}^2+(y-0{)}^2=100$
3. $x^2+y^2=100$
4. $(x+6{)}^2+(y+8{)}^2=100$

Q. If $N$ is the normal to the circle $x^2+y^2+4x+6y-39=0$ at the point $(2,3)$, then it meets the circle again at

1. $(2,9)$
2. $(-6,3)$
3. $(2,-9)$
4. $(-6,-9)$

Q. Let the tangent drawn at $(-1,2)$ to the circle $x^2+y^2-3x-3y-2=0$ is normal to the circle $x^2+y^2-2ay+b=0.$ If the radius $\left(r\right)$ of the second circle is such that $\left[r\right]=1,$ then
([.] denotes the greatest integer function)

1. $b\in [48,49)$
2. $b\in (45,48]$
3. $b\in [45,48)$
4. $b\in (48,49]$

Q. The line $2x-y+1=0$ is tangent to a circle at point $(2,5)$ and the centre of the circle lies on $x-2y=4.$ Then the radius of the circle is ____ units.

1. $3\sqrt{5}$
2. $5\sqrt{3}$
3. $2\sqrt{5}$
4. $5\sqrt{2}$

Q. If the tangents drawn from point $P(-4,0)$ to the circle $x^2+y^2=4$ touches the circle at points $Q$ and $R,$ then $QR=$

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

Q.

Find the equation of the normal to the circle $x^2+y^2-6x-8y-24=0$ at the point (4, 6).

1. 2x -y = 2

2. 3x + y = 18

3. 3x - y = 12

4. -3y + 12 = 0

Q.

If in general quadratic equation f(x, y) = 0, $△=0$

and a + b = 0, then the equation represents

1. None of these

2. Two lines passing through origin

3. Two parallel straight lines

4. Two perpendicular straight lines

Q. Given $c_1:x^2+y^2-2x-2y+3=0$ and $c_2:x^2-y^2-2x-2y+7=0$ are

1. none of these
2. concentric circles
3. congruent circles
4. similar circles

Q. Let ABCD be a quadrilateral in which $AB\parallel CD,AB\perp AD$ and AB=3CD. The area of quadrilateral ABCD is 4. The radius of a circle touching all the sides of quadrilateral is:

1. $sin\frac{\pi }{12}$
2. $sin\frac{\pi }{6}$
3. $sin\frac{\pi }{4}$
4. $sin\frac{\pi }{3}$

Q. The equation of the normal to the circle $x^2+y^2-2x=0$ parallel to the line $x+2y=3$ is

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

Q. The line 4x-3y=-12 is the tangent at point A(-3, 0) and the line 3x+4y=16 is the tangent at the point B(4, 1) to a circle. The equation of circle is .

1. $(x+1{)}^2+(y+3{)}^2=25$
2. $(x+1{)}^2+(y-3{)}^2=25$
3. $(x-1{)}^2+(y+3{)}^2=25$
4. $(x-1{)}^2+(y-3{)}^2=25$

Q. In the Fig., two circles touch each other internally at a point A. The radius of the smaller circle with centre M is 5cm. The smaller circle passes through the centre N of the larger circle. The tangent to the smaller circle drawn through C intersects the larger circle in point D. Then CD is

1. $\frac{80\sqrt{2}}{3}$
2. $\frac{10\sqrt{2}}{3}$
3. $\frac{20\sqrt{2}}{3}$
4. $\frac{40\sqrt{2}}{3}$

Q. The point P (2, 4) lies on a circle of radius 6 and centre C (3, 5).
Enter $1$ for true and $0$ for false.

Q.

The normal to the circle $x^2+y^2+2x-10y+k$ = 0 which is perpendicular to x-3y+2=0 is

1. 3x+y=0

2. 3x+y=2

3. x+3y+2=0

4. 3x+y=4

Q. The equation of the normals to the circle $x^2+y^2-8x-2y+12=0$ at the points whose ordinate is $-1$ is/are

1. $2x+y-9=0$
2. $2x-y-7=0$
3. $2x+y-7=0$
4. $2x-y-9=0$

Q. If the line $ax+by=2$ is a normal to the circle $x^2+y^2-4x-4y=0$ and a tangent to the circle $x^2+y^2=1,$ then

1. $a=\frac{1}{2},b=\frac{1}{2}$
2. $a=\frac{1-\sqrt{7}}{2},b=\frac{1+\sqrt{7}}{2}$
3. $a=\frac{1}{4},b=\frac{3}{4}$
4. $a=1,b=2$

Q.

If the equation of normal to the circle $x^2+y^2-16x-12y+99=0$, which is also a tangent to $x^2+y^2=4$ is ax + by + c = 0, find the value of $\left[|\right(\frac{a}{b}\left)|\right]$, where [x] is the greatest integer function

1. 0

2. 1

3. 2

4. 4

Q. If $N$ is the normal to the circle $x^2+y^2+4x+6y-39=0$ at the point $(2,3)$, then it meets the circle again at

1. $(2,9)$
2. $(-6,3)$
3. $(2,-9)$
4. $(-6,-9)$

Q.

The normal to the circle $x^2+y^2+2x-10y+k$ = 0 which is perpendicular to x-3y+2=0 is

1. x+3y+2=0

2. 3x+y=0

3. 3x+y=2

4. 3x+y=4

Q. Show that the four points P, Q, R, S with position vectors , , , respectively such that 5−2+6−9=, are coplanar. Also, find the position vector of the point of intersection of the line segments PR and QS.