Length of Tangent and Power of a Point

Q.

If the line $y=mx+c$ is a common tangent to the hyperbola $\left(\frac{x^2}{100}\right)-\left(\frac{y^2}{64}\right)=1$ and the circle $x^2+y^2=36$, then which one of the following is true?

1. $4c^2=369$

2. $c^2=369$

3. $8m+5=0$

4. $5m=4$

Q. Find the ratio of the length of the tangents from any point on the circle $15x^2+15y^2-48x+64y=0$ to the two circles $5x^2+5y^2-24x+32y+75=0,$ $5x^2+5y^2-48x+64y+300=0.$

1. $1:2$
2. $2:5$
3. $1:3$
4. $1:4$

Q.

C is a circle centered at O. $P{T}_1$ and $P{T}_2$ are the tangents drawn from a point P outside the circle to the circle. Then which of these is the most appropriate relation.

1. $OP>P{T}_2=P{T}_1$

2. $OP=P{T}_1=P{T}_2$

3. $OP<P{T}_1\le P{T}_2$

4. $OP<P{T}_1=P{T}_2$

Q. If the tangent at the point $P$ on the circle $x^2+y^2+6x+6y=2$ meets the straight line $5x-2y+6=0$ at point $Q$ on the $y$-axis, then the length of $PQ$ is

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

Q. Find the ratio of the length of the tangents from any point on the circle $15x^2+15y^2-48x+64y=0$ to the two circles $5x^2+5y^2-24x+32y+75=0,$ $5x^2+5y^2-48x+64y+300=0.$

1. $1:2$
2. $2:5$
3. $1:3$
4. $1:4$

Q. With usual notation, if in a triangle $ABC,$ $a+3=b+2=11$ and $\cos C=\frac{2}{3},$ then

1. Length of the tangent from the vertex $C$ to the circle escribed to side $AB$ is $12.$
2. Length of the tangent from the vertex $C$ to the incircle of the triangle is $4.$
3. Sum of lengths of tangents from the vertices $A,B,C$ to the circle escribed to the side $AB$ is $38.$
4. Sum of lengths of tangents from the vertices $A,B,C$ to the incircle of the triangle is $24.$

Q. The length of tangent drawn from the point $(1,1)$ to the circle $x^2+y^2+8x+20y+70=0$ is units.

Q.

Find the length of the tangent from a point (6, 1) to the circle $x^2+y^2-4x=0.$

1. 5

2. 4

3. $\sqrt{17}$

4. $\sqrt{13}$