Length of External Common Tangent

Q.

A pair of tangents are drawn from the origin to the circle $x^2+y^2+20(x+y)+20=0$. The equation of the pair of tangents is

1. 

2. 

3. 

4. 

Q.

The equation of the common tangent to $x^2+y^2=2a^2$ and $y^2=8ax$ is

1. $y=\pm (x+a)$
2. $y=\pm (x+2a)$
3. $y=\pm (x+3a)$.
4. $y=\pm (x+4a)$.

Q. If two different tangents of $y^2=4ax$ are normals to $x^2=4y$, then

1. $|a|>2\sqrt{2}$
2. $|a|<2\sqrt{2}$
3. $|a|>\frac{1}{2\sqrt{2}}$
4. $|a|<\frac{1}{2\sqrt{2}}$

Q. If $y={\sin }^{-1}x$, prove that $(1-x^2)y_2-x{y}_1=0$

Q. Find the area of triangle formed by the tangents to the curve $2y^2=x$ drawn at the points whose ordinates are 2, 3, 4 respectively

1. 1 sq.units
2. 24 sq units
3. 12 sq units
4. 9 sq units

Q. Let $A$ be the centre of the circle $x^2+y^2–2x–4y–20=0.$ Let $B(1,7)$ and $D(4,–2)$ be two points on the circle such that tangents at $B$ and $D$ meet at $C$. The area of the quadrilateral $ABCD$ is

1. $150\text{sq. units}$
2. $50\text{sq. units}$
3. $75\text{sq. units}$
4. $70\text{sq. units}$

Q. If $y=(si{n}^{-1}x{)}^2$, show that $(1-x^2)\frac{d^{2}y}{d{x}^2}-x\frac{dy}{dx}=2$.

Q. If the lengths of the tangents from two points A, B to a circle are 6, 7 respectively. If A, B are conjugate points then AB =

1. 5
2. $\sqrt{85}$
3. $\frac{\sqrt{85}}{2}$
4. none

Q.

Find the length of direct common tangent for circles $x^2+y^2-20x+64=0$and $x^2+y^2+30x+144=0$

1. $2\sqrt{150}$

2. 20

3. $2\sqrt{154}$

4. 9

Q.

Find the length of direct common tangent for circles $x^2+y^2-20x+64=0$and $x^2+y^2+30x+144=0$

1. 20

2. 9

3. 

4. 

Q.

If one circle lie completely outside the other circle, Q , the intersection of direct common tangent, lie on the line joining the centers of the two circles and divide it internally in the ratio of their radii.

1. True

2. False

Q. If the lengths of the tangents from two points A, B to a circle are 6, 7 respectively. If A, B are conjugate points then AB =

1. 
2. 
3. none
4. 5

Q. If $y={\sin }^{-1}[x\sqrt{x-1}-\sqrt{x}\sqrt{1-x^2}]$, find $\frac{dy}{dx}$.

Q.

If one circle lie completely outside the other circle, Q , the intersection of direct common tangent, lie on the line joining the centers of the two circles and divide it internally in the ratio of their radii.

1. True

2. False

Q.

The two circles $x^2+y^2+2ax+c=0and{x}^2+y^2+2by+c=0$ touch if $\frac{1}{a^2}+\frac{1}{b^2}=$

1. 

2. 

3. 

4. 

Q. Find the value of $a$, if the distance between the points $A(3,14)$ and $B(a,5)$ is $9$ units.

Q.

On the circle with centre O, points A, B are such that OA = AB. A point C is located on the tangent at B to the circle such that A and C are on the opposite sides of the line OB and AB = BC. The line segment AC intersects the circle again at F. Then the ratio $\angle$BOF: $\angle$BOC is equal to:

1. 2:3

2. 3:4

3. 1:2

4. 4:5

Q.

Find the length of direct common tangent for circles $x^2+y^2-20x+64=0$and $x^2+y^2+30x+144=0$

1. $2\sqrt{150}$

2. 20

3. $2\sqrt{154}$

4. 9