Length of Tangent and Power of a Point

Q. Let $L_1,L_2$ and $L_3$ be the lengths of tangents drawn from a point $P(h,k)$ to the circles $x^2+y^2=4,$ $x^2+y^2–4x=0$ and $x^2+y^2–4y=0$ respectively. Let $L_1^4=L_2^2{L}_3^2+16$ gives the locus of $P$ as two curves, $C_1$ (straight line) and $C_2$ (circle). A triangle is formed by $C_1$ and two other lines which are at an angle of ${45}^{\circ }$ with $C_1$ and tangent to the circle $C_2.$ Then which of the following is/are correct?

1. Straight line which intersects both the curves $C_1$ and $C_2$ orthogonally, is $x–y=0$.
2. Straight line which intersects both the curves $C_1$ and $C_2$ orthogonally, is $x+y=0$.
3. Circumcenter of the triangle is at origin.
4. Circumcenter of the triangle is at $(1,1).$

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

Q.

A tangent is drawn to a circle of radius r from an external point P .If length of tangent from a point to the circle is twice the minimum distance between circle and the point, what is the length of the tangent?

1. 

2. 

3. 

4. 3 r

Q. As shown in figure, three circles which have the same radius $r$, have centres at $(0,0),(1,1)$ and $(2,1)$. If they have a common tangent line, as shown, then the value of radius is

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

Q. As shown in figure, three circles which have the same radius $r$, have centres at $(0,0),(1,1)$ and $(2,1)$. If they have a common tangent line, as shown, then the value of radius is

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

Q. Let tangents are drawn from $P(3,4)$ to the circle $x^2+y^2=a^2$ touches the circle at $A$ and $B$. If area of $△PAB=\frac{192}{25}$ sq. units, then $a=$

Q. ​​​​​$\begin{array}{cccc}& \text{List 1}& & \text{List II}\\ \text{(1)}& \text{If PQ is a focal chord of the ellipse}& & \\ & \frac{x^2}{25}+\frac{y^2}{16}=1\text{which passes through S(3, 0) \&}& & \\ & \text{PS=2, then the length of the focal chord PQ is}& \left(P\right)& 1\\ \text{(2)}& \text{The focal chord of}{y}^2=16x\text{is tangent to}& & \\ & (x-6{)}^2+y^2=2\text{. Then the possible values}& & \\ & \text{of the slopes of the chord is}& \left(Q\right)& 3\\ \text{(3)}& \text{A circle is described on the focal chord of}& & \\ & y^2=-12x\text{as a diameter such that it touches}& & \\ & \text{the line}x=a\text{. Then the value of}a\text{is}& \left(R\right)& 6\\ \text{(4)}& \text{If the eccentricity of the hyperbola is}\frac{5}{4}and& & \\ & \text{2x+3y-10=0 is a focal chord of the hyperbola}\frac{x^2}{a^2}-\frac{y^2}{b^2}=1,& & \\ & \text{then the length of transverse axis is}& \left(S\right)& 8\\ & & \left(T\right)& 10\\ & & & \\ & & \left(U\right)& 4\end{array}$
Which of the following is only incorrect combination?

1. $1\to \left(T\right)$
2. $2\to \left(P\right)$
3. $3\to \left(Q\right)$
4. $4\to \left(R\right)$

Q. 37. The tangent frm the point of intersection of line 2x-3y+1=0 aand 3x-2y-1=0 to the circle xsquare+ysquare+2x-4y=0 is

Q. If an incident ray from the point $(-1,2)$ parallel to the axis of the parabola
$y^2=4x$ strikes the parabola, then what is the equation of the reflected ray?

Q. If a chord is drwan through any point $P$ of the circle $x^2+y^2+4x-6y-12=0$ become secant for circle $x^2+y^2+4x-6y+4=0$ intersecting it at points at $A$ and $B$, then $PA\cdot PB$ is

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. A ray of light is moving along the line $y=x+2,$ gets reflected from a parabolic mirror whose equation is
$y^2=4(x+2).$ The reflection does not happen at the vertex of the parabola. After reflection, the ray does not pass through the focus of the parabola. What will be the equation of the line which contains the reflected ray?

1. $x+7y+10=0$
2. $x-7y+26=0$
3. $x-7y-10=0$
4. $x+7y-10=0$

Q. The length of chord of circle $x^2+y^2+6x+9y+8=0$ intercepted by $x$ axis is

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

Q. Let tangents are drawn from $P(3,4)$ to the circle $x^2+y^2=a^2$ touches the circle at $A$ and $B$. If area of $△PAB=\frac{192}{25}$ sq. units, then $a=$

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. 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 the common chord of the circles $(x-a{)}^2+(y-b{)}^2=c^2$ and $(x-b{)}^2+(y-a{)}^2=c^2$ is

1. $\sqrt{4c^2-2(a-b{)}^2}$
2. $\sqrt{2c^2-(a-b{)}^2}$
3. $\sqrt{4c^2-(a-b{)}^2}$
4. $\sqrt{4c^2+2(a-b{)}^2}$

Q. Consider the circle $x^2+y^2=9$ and the parabola $y^2=8x$. They intersect at $P$ and $Q$ in the first and fourth quadrants, respectively. Tangents to the circle at $P$ and $Q$ intersect the $x$-axis at $R$ and tangent to the parabola at $P$ and $Q$ intersect the $x$-axis at $S$.
The radius of the incircle of the triangle $PQR$ is -

1. $4$
2. $3$
3. $\frac{8}{3}$
4. $2$

Q.

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

1. 

2. 

3. 5

4. 4

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 equation of common tangents of x²+y²=6 and y²=8(3)^1/2x

1. y=x+23
2. y=x+3
3. x+y+23=0
4. x-y-3=0

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. 

2. 

3. 

4. 

Q. $x^2+y^2=a^2$ and $(x-c{)}^2+y^2=b^2$ are two intersecting circles. lf $a,b,c$ are the sides $BC,CA,AB$ of $\Delta ABC$. lf $p_1,p_2,p_3$ are the altitudes through $A,B,C$ respectively then the length of the common chord is

1. $2p_1$
2. $2p_2$
3. $2p_3$
4. $p_1$

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. If the length of the tangents from P(1, -1) and Q(3, 3) to a circle are $\sqrt{2}\mathrm{and}\sqrt{6}$ then the length of a tangent from R(-1, -5) to the same circle is .

1. $\sqrt{37}$
2. $\sqrt{38}$
3. $\sqrt{35}$
4. $\sqrt{36}$

Q. $p_1$ and $p_2$ are the distances of a variable point from the
two lines $2x+y-1=0$ and $x+2y+1=0$ satisfying the
condition that $p_1^2=p_2^2+3$. Prove that the product of
the distance of the same point from the lines
$x+y+2=0$ and $x-y-3=0$ varies as its distance from the line $x-3y-1=0$.

Q. Length of tangent drawn from any point of circle $x^2+y^2+2gx+2fy+c=0$ to the circle $x^2+y^2+2gx+2fy+d=0$, (d > c) is

1. $\sqrt{c-d}$
2. $\sqrt{d-c}$
3. $\sqrt{g-f}$
4. $\sqrt{f-g}$

Q. The common tangents to the circle $x^2+y^2=2$ and the parabola $y^2=8x$ touch the circle at the points $P,Q$ and the parabola at the points $R,S$. Then the area of the quadrilateral $PQRS$ is

1. $3$
2. $6$
3. $9$
4. $15$

Q. The length of the common chord of the circles: $x^2+y^2+6x+5=0$ and $x^2+y^2+4y-5=0$ is

1. $\frac{\sqrt{12}}{13}$
2. $\sqrt{\frac{12}{13}}$
3. $\frac{12}{\sqrt{13}}$
4. $\sqrt{\frac{13}{12}}$

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. $1:4$