Tangent

Q.

The equation of the circle passing through the foci of the ellipse $\left(\frac{x^2}{16}\right)+\left(\frac{y^2}{9}\right)=1$ and having centre at $\left(0,3\right)$ is

1. $x^2+y^2-6y-7=0$

2. $x^2+y^2-6y+7=0$

3. $x^2+y^2-6y-5=0$

4. $x^2+y^2-6y+5=0$

Q.

The perpendicular distance of center from tangent of the circle is equal to its radius.

1. True

2. False

Q. The equation of the circle passing through the point (1, –2) and having its centre on the line 2x – y – 14 = 0 and touching the line 4x + 3y – 23 = 0

1. $x^2$ + $y^2$ + 8x + 12y + 27 = 0
2. $x^2$ + $y^2$ - 12y + 27 = 0
3. $x^2$ + $y^2$ - 8x + 12y - 27 = 0
4. $x^2$ + $y^2$ - 8x + 12y + 27 = 0

Q.

The foot of the perpendicular of center on a tangent and the point of contact of the tangent are same.

1. True

2. False

Q. If a circle whose centre is (1, –3) touches the line 3x – 4y –5 = 0, then the radius of the circle is

1. 2
2. 4
3. $\frac{5}{2}$
4. $\frac{7}{2}$

Q. Tangents $PA$ and $PB$ are drawn to the circle $(x-4{)}^2+(y-5{)}^2=4$ from the point $P$ on the curve $y=\sin x$, where $A$ and $B$ lie on the circle. Consider the function y = $f\left(x\right)$ represented by the locus of the centre of the circumcircle of triangle $PAB$, then
Range of $y=f\left(x\right)$ is

1. $[-2,1]$
2. $[-1,4]$
3. $[0,2]$
4. $[2,3]$

Q. If the line lx + my = 1 be a tangent to the circle $x^2+y^2=a^2,$ then the locus of the point (l, m) is

1. A straight line

2. A Circle

3. A parabola
4. An ellipse

Q. Consider the relation $4l^2-5m^2+6l+1=0$, where $l,m\in R$, then the line $lx+my+1=0$ touches a fixed circle whose centre and radius of circle are

1. $(2,0),3$
2. $(-3,0),\sqrt{3}$
3. $(3,0),\sqrt{5}$
4. $(-2,0),3$

Q. Solve the equation $24x^3-14x^2-63x+45=0$, one root being double another.

Q. The equation of circle which touches the axes of coordinates and the line $fracx3+\frac{y}{4}$ and whose centre lies in the first quadrant is $x^2+y^2-2cx-2cy+c^2=0,$ where c is

1. 1
2. 2
3. 3
4. 6

Q. The equation of circle with centre (1, 2) and tangent x + y - 5 = 0 is

1. $x^2+y^2+2x-4y+6=0$
2. $x^2+y^2-2x-4y+3=0$
3. $x^2+y^2-2x+4y+8=0$
4. $x^2+y^2-2x-4y+8=0$

Q. A circle is tangent to the x and y axes in the first quadrant at the points $P$ and $Q$ respectively. $BC$ and $AD$ are parallel tangents to the circle with slope $-1$. If the points $A$ and $B$ are on the y- axis while $C$ and $D$ are on the x-axis and the area of the quadrilateral $ABCD$ is $900\sqrt{2}$ sq. units, then the radius of the circle is

1. $10$
2. $12$
3. $15$
4. $20$

Q. The equation to the circle with centre (2, 1) and touching the line 3x + 4y = 5 is

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