Angle between Pair of Tangents

Q. The angle between a pair of tangents drawn from a point $P$ to the circle $x^2+y^2+4x-6y+9{\sin }^2\alpha +13{\cos }^2\alpha =0$ is $2\alpha .$ The equation of the locus of the point $P$ is

1. $x^2+y^2+4x+6y+9=0$
2. $x^2+y^2-4x+6y+9=0$
3. $x^2+y^2–4x–6y+9=0$
4. $x^2+y^2+4x-6y+9=0$

Q.

Two tangents are drawn from a point$P$ to the circle $x^2+y^2–2x–4y+4=0,$such that the angle between these tangents is ${\tan }^{-1}\left(\frac{12}{5}\right),$where ${\tan }^{-1}\left(\frac{12}{5}\right)\in \left(0,\mathrm{\pi }\right)$ If the centre of the circle is denoted by $C$and these tangents touch the circle at points $A$ and $B,$ then the ratio of the areas of $△PAB$ and $△CAB$ is:

1. $11:4$

2. $9:4$

3. $2:1$

4. $3:1$

Q. Consider the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$. The portion of the tangent at any point of the curve intercepted between the point of contact and the directrix subtends at the corresponding focus an angle of

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

Q. The range of values of $a$ such that the angle $\theta$ between the pair of tangents drawn from $(a,0)$ to the circle $x^2+y^2=1$ satisfies $\frac{\pi }{2}<\theta <\pi$, lies in

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

Q.

The angle between the tangents drawn at the points $(5,12)$and $(12,-5)$ to the circles $x^2+y^2=169$ is

1. ${45}^{°}$

2. ${60}^{°}$

3. ${30}^{°}$

4. ${90}^{°}$

Q.

The cosine of the angle between the tangents from the origin to the circle $x^2+y^2-14x+2y+25$ = 0 is

1. 1

2. 0

3. -1/2

4. 

Q. The product of slope of tangents from point $(0,1)$ to the circle $x^2+y^2-2x+4y=0$ is

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

Q. The angle between a pair of tangents drawn from a point P to the circle x${}^2$ + y${}^2$ + 4x - 6y + 9 sin${}^2$$\alpha$ + 13 cos${}^2$$\alpha$ = 0 is 2$\alpha$. The equation of the locus of the point P is

1. 
2. 
3. 
4. 

Q. A normal to the hyperbola, $4x^2-9y^2=36$ meets the co-ordinate axes $x$ and $y$ at $A$ and $B$, respectively. If the parallelogram $OABP$ ($O$ being the origin) is formed, then the locus of $P$ is :

1. $4x^2-9y^2=121$
2. $4x^2+9y^2=121$
3. $9x^2-4y^2=169$
4. $9x^2+4y^2=169$

Q. The angle between the pair of tangents drawn from a point P to the circle $x^2+y^2+4x-6y+9$$si{n}^2\alpha +13co{s}^2\alpha =0$ is $2\alpha$. Then the equation of the locus of the point P is

1. $x^2+y^2+4x-6y+4=0$
2. $x^2+y^2+4x-6y-9=0$
3. $x^2+y^2+4x-6y-4=0$
4. $x^2+y^2+4x-6y+9=0$

Q. The range of values of $a$ such that the angle $\theta$ between the pair of tangents drawn from $(a,0)$ to the circle $x^2+y^2=1$ satisfies $\frac{\pi }{2}<\theta <\pi$, lies in

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

Q. Angle between tangents drawn to $x^2+y^2-2x-4y+1=0$ at the points where it is cut by the line $y=2x+c,is\frac{\pi }{2}$ then

1. $|c|=\sqrt{5}$
2. $|c|=2\sqrt{5}$
3. $|c|=\sqrt{10}$
4. $|c|=2\sqrt{10}$

Q. Common tangents are drawn to parabola $y^2=4x$ and ellipse $3x^2+8y^2=48$ touching the parabola at A and B and the ellipse at C and D. Area of quadrilateral ABCD is ?

1. $55\sqrt{2}squnits$
2. $44\sqrt{2}squnits$
3. $55squnits$
4. $44squnits$

Q. Let $2x^2+y^2-3xy=0$ be the equation of a pair of tangents drawn from the origin $O$ to a circle of radius $3$ units with centre in the first quadrant. If $A$ is one of the points of contact, then the length of $OA$ is

1. $9(1+\sqrt{10})$ units
2. $3(3+\sqrt{10})$ units
3. $9+\sqrt{10}$ units
4. $3+\sqrt{10}$ units

Q. Let $S$ be a circle whose centre is $C(2,3)$ and touching the $y$-axis. Tangents $OA$ and $OB$ are drawn from the origin $O$ to the circle which touches the circle at $A$ and $B.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The finite slope of the tangent OA is}& \text{(P)}& \frac{12}{13}\\ \text{(B)}& \text{If}\alpha \text{is the acute angle between the}& \text{(Q)}& \frac{54}{13}\\ & \text{tangents, then the value of}\sin \alpha \text{is}& & \\ \text{(C)}& \text{If}P(x_1,y_1)\text{is any point on the circle,}& \text{(R)}& \frac{5}{12}\\ & \text{then the product of the minimum and the}& & \\ & \text{maximum values of}OP\text{, is}& & \\ \text{(D)}& \text{A line is drawn from}S(4,5)\text{, intersecting}& \text{(S)}& 9\\ & \text{the circle at}M\text{and}N\text{. Then the value of}& & \\ & SM\cdot SN\text{is}& & \\ & & \text{(T)}& 4\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(A)}\to \text{(R)},\text{(B)}\to \text{(P)}$
2. $\text{(A)}\to \text{(P)},\text{(B)}\to \text{(R)}$
3. $\text{(A)}\to \text{(R)},\text{(B)}\to \text{(S)}$
4. $\text{(A)}\to \text{(Q)},\text{(B)}\to \text{(P)}$

Q.

The angle between a pair of tangents drawn from a point P to the circle $x^2+y^2+4x-6y+9si{n}^2\alpha +13co{s}^2\alpha =0is2\alpha .$ The equation of the locus of the point P is

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

2. $x^2+y^2+4x-6y-9=0$
   

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

4. $x^2+y^2+4x-6y+9=0$

Q. The slope of the normal to the curve $y=2x^2+3\sin x$ at $x=0$ is.

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

Q. Angle between tangents drawn to $x^2+y^2-2x-4y+1=0$ at the points where it is cut by the line $y=2x+c,is\frac{\pi }{2}$ then

1. 
2. 
3. 
4. 

Q. The angle between the tangents from $(\alpha ,\beta )$ to the circle $x^2+y^2=a^2$, is

1. $Noneofthese$
2. $ta{n}^{-1}\frac{a}{\sqrt{{\alpha }^2+{\beta }^2-a^2}}$
3. $ta{n}^{-1}\frac{\sqrt{{\alpha }^2+{\beta }^2-a^2}}{a}$
4. $2ta{n}^{-1}\frac{a}{\sqrt{{\alpha }^2+{\beta }^2-a^2}}$

Q. A point on the curve $f\left(x\right)=\sqrt{x^2-4}$ defined in $[2,4]$ where the tangent is parallel to the chord joining two points on the curve

1. $(\sqrt{2},\sqrt{6})$
2. $(\sqrt{6},\sqrt{2})$
3. $(6,2)$
4. $(2,6)$

Q.

The angle between a pair of tangents drawn from a point P to the circle $x^2+y^2+4x-6y+9si{n}^2\alpha +13co{s}^2\alpha =0is2\alpha .$ The equation of the locus of the point P is

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

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

3. $x^2+y^2+4x-6y-9=0$
   

4. $x^2+y^2+4x-6y+9=0$

Q. The angle between the pair of tangents drawn from a point P to the circle $x^2+y^2+4x-6y+9$$si{n}^2\alpha +13co{s}^2\alpha =0$ is $2\alpha$. Then the equation of the locus of the point P is

1. $x^2+y^2+4x-6y+4=0$
2. $x^2+y^2+4x-6y-9=0$
3. $x^2+y^2+4x-6y-4=0$
4. $x^2+y^2+4x-6y+9=0$

Q. Let $S$ be a circle whose centre is $C(2,3)$ and touching the $y$-axis. Tangents $OA$ and $OB$ are drawn from the origin $O$ to the circle which touches the circle at $A$ and $B.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The finite slope of the tangent OA is}& \text{(P)}& \frac{12}{13}\\ \text{(B)}& \text{If}\alpha \text{is the acute angle between the}& \text{(Q)}& \frac{54}{13}\\ & \text{tangents, then the value of}\sin \alpha \text{is}& & \\ \text{(C)}& \text{If}P(x_1,y_1)\text{is any point on the circle,}& \text{(R)}& \frac{5}{12}\\ & \text{then the product of the minimum and the}& & \\ & \text{maximum values of}OP\text{, is}& & \\ \text{(D)}& \text{A line is drawn from}S(4,5)\text{, intersecting}& \text{(S)}& 9\\ & \text{the circle at}M\text{and}N\text{. Then the value of}& & \\ & SM\cdot SN\text{is}& & \\ & & \text{(T)}& 4\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(T)},\text{(D)}\to \text{(S)}$
2. $\text{(C)}\to \text{(S)},\text{(D)}\to \text{(T)}$
3. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(Q)}$
4. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(T)}$

Q. Find the equation of tangent and normal at t, to the curve $x=asi{n}^{3}t,y=aco{s}^{3}t$

Q. Let $S$ be a circle whose centre is $C(2,3)$ and touching the $y$-axis. Tangents $OA$ and $OB$ are drawn from the origin $O$ to the circle which touches the circle at $A$ and $B.$

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{The finite slope of the tangent OA is}& \text{(P)}& \frac{12}{13}\\ \text{(B)}& \text{If}\alpha \text{is the acute angle between the}& \text{(Q)}& \frac{54}{13}\\ & \text{tangents, then the value of}\sin \alpha \text{is}& & \\ \text{(C)}& \text{If}P(x_1,y_1)\text{is any point on the circle,}& \text{(R)}& \frac{5}{12}\\ & \text{then the product of the minimum and the}& & \\ & \text{maximum values of}OP\text{, is}& & \\ \text{(D)}& \text{A line is drawn from}S(4,5)\text{, intersecting}& \text{(S)}& 9\\ & \text{the circle at}M\text{and}N\text{. Then the value of}& & \\ & SM\cdot SN\text{is}& & \\ & & \text{(T)}& 4\end{array}$

Which of the following is the only CORRECT combination?

1. $\text{(C)}\to \text{(T)},\text{(D)}\to \text{(S)}$
2. $\text{(C)}\to \text{(S)},\text{(D)}\to \text{(T)}$
3. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(T)}$
4. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(Q)}$

Q. Find the equations of the tangent and normal to the given curves at the indicated points:
(i) $y=x^4-6x^3+13x^2-10x+5$ at $(0,5)$.
(ii) $y=x^4-6x^3+13x^2-10x+5$ at $(1,3)$
(iii) $y=x^3$ at $(1,1)$
(iv) $y=x^2$ at $(0,0)$
(v) $x=\cos t,y=\sin t$ at $t=\frac{\pi }{4}$

Q. If the tangent at $\left(\theta \right)$ to the circle $x^2+y^2=4$ touches the circle $x^2+y^2-6\sqrt{3}x-6y+20=0$ then one of the values of $\theta$ is?

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

Q. Angle between tangents drawn to $x^2+y^2-2x-4y+1=0$ at the points where it is cut by the line $y=2x+c,is\frac{\pi }{2}$ then

1. $|c|=\sqrt{5}$
2. $|c|=2\sqrt{5}$
3. $|c|=\sqrt{10}$
4. $|c|=2\sqrt{10}$

Q. The angle between the two tangents from the origin to the circle $(x-7{)}^2+(y+1{)}^2=25$ is

1. $\frac{\pi }{3}$
2. $\frac{\pi }{6}$
3. $\frac{\pi }{2}$
4. 0

Q. The angle between the tangents from a point on $x^2+y^2+2x+4y-31=0$ to the circle $x^2+y^2+2x+4y-4=0$ is

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