Pair of Tangents from an External Point

Q. Angle between the tangents drawn from the origin to the parabola $y^2=4a(x-a)$ is

1. ${30}^{\circ }$
2. ${45}^{\circ }$
3. ${60}^{\circ }$
4. ${90}^{\circ }$

Q. Let the tangent to the parabola $S:y^2=2x$ at the point $P(2,2)$ meet the $x-$axis at $Q$ and normal at it meet the parabola $S$ at the point $R.$ Then the area (in sq. units) of the triangle $PQR$ is equal to

1. $\frac{25}{2}$
2. $\frac{15}{2}$
3. $\frac{35}{2}$
4. $25$

Q. $PG$ is the normal at $P$ to the parabola $y^2=4ax$. $G$ is on the axis, $GP$ is produced to $Q$ such that $PQ=GP$, then

1. Locus of $Q$ is $y^2=16a(x+2a)$
2. Focus of locus of $Q$ is $(2a,0)$
3. Latus rectum of locus of $Q$ is $16a$
4. Locus of $Q$ is $y^2=16a(x-2a)$

Q. The length of the chord of the parabola $x^2=4y$ having equation $x-\sqrt{2}y+4\sqrt{2}=0$ is :

1. $3\sqrt{2}$
2. $2\sqrt{11}$
3. $6\sqrt{3}$
4. $8\sqrt{2}$

Q.

Find the equation of the tangent line to the curve $y=x^2-2x+7$ which is

(a) parallel to the line 2x-y+9=0.

(a) parallel to the line 5y-15x=13.

Q. Suppose that the foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{5}=1$ are $(f_1,0)$ and $(f_2,0)$ where $f_1>0$ and $f_2<0$. Let $P_1$ and $P_2$ be two parabolas with a common vertex at $(0,0)$ and with foci at $(f_1,0)$ and $(2f_2,0)$, respectively. Let $T_1$ be a tangent to $P_1$ which passes through $(2f_2,0)$ and $T_2$ be a tangent to $P_2$ which passes through $(f_1,0)$. If $m_1$ is the slope of $T_1$ and $m_2$ is the slope of $T_2$, then the value of $(\frac{1}{m_1^2}+m_2^2)$ is .

Q.

Tangents are drawn from the point $(-1,2)$ to the parabola $y^2=4x$. The length of the intercept made by the line $x=2$ on these tangents is

1. $6$
2. $6\sqrt{2}$
3. $2\sqrt{6}$
4. None

Q. If the locus of mid point of the chords of the parabola $y^2=4ax$ which passes through a fixed point $(h,k)$ is also a parabola, then length of its latus rectum (in units) is

1. $4a$
2. $8a$
3. $6a$
4. $2a$

Q. If two tangents drawn from the point $(\alpha ,\beta )$ to the parabola $y^2=4x$ such that the slope of one tangent is double of the other, then

1. $\beta =\frac{2}{9}{\alpha }^2$
2. $\alpha =\frac{2}{9}{\beta }^2$
3. $2\alpha =9{\beta }^2$
4. $\alpha =2{\beta }^2$

Q. If the tangents at $P$ and $Q$ on the parabola $y^2=4ax$ meets at $T$ and if $S$ is the focus of the parabola then, $SP,ST,SQ$ are in

1. A.P.
2. G.P.
3. H.P
4. A.G.P.

Q. If the locus of mid point of any normal chord of the parabola $y^2=4x$ is $x-a=\frac{b}{y^2}+\frac{y^2}{c};$ where $a,b,c\in N$, then $(a+b+c)$ is equal to

Q. If the area of the quadrilateral formed by the tangents from the origin to the circle $x^2+y^2+6x-10y+c=0$ and radii corresponding to the point of contact is $15$ sq. units, then value(s) of $c$ can be

1. $5$
2. $9$
3. $16$
4. $25$

Q. Consider the hyperbola $H:x^2-y^2=1$ and a circle $S$ with center $N(x_2,0).$ Suppose that $H$ and $S$ touch each other at a point $P(x_1,y_1)\text{with}{x}_1>1\text{and}{y}_1>0.$ The common tangent to $H$ and $S$ at $P$ intersects the $x-$axis at point $M$. If $(l,m)$ is the centroid of the triangle $△PMN,$ then the correct expression(s) is (are)

1. $\frac{dl}{d{x}_1}=1-\frac{1}{3x_1^2}\text{for}{x}_1>1$
2. $\frac{dm}{d{x}_1}=\frac{x_1}{3(\sqrt{x_1^2-1})}\text{for}{x}_1>1$
3. $\frac{dl}{d{x}_1}=1+\frac{1}{3x_1^2}\text{for}{x}_1>1$
4. $\frac{dm}{d{y}_1}=\frac{1}{3}\text{for}{y}_1>1$

Q. Equation(s) of the tangents drawn from $(4,10)$ to the parabola $y^2=9x$ is/are

1. $x-4y+36=0$
2. $9x-4y+4=0$
3. $x-2y+4=0$
4. $x-y+1=0$

Q. The area of the triangle formed by any tangent to the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ with its asymptotes is

1. 
2. 
3. 
4. 

Q. If $\theta$ is the angle between the two tangents to $y^2=12x$ drawn from the point $(1,4)$, then $\tan \theta$ is equal to

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

Q.

Number of points from where perpendicular tangents to the curve $\frac{x^2}{16}-\frac{y^2}{25}=1$ can be drawn, is:

1. 2

2. 1

3. 3

4. 0

Q. The base of triangle is divided into three equal parts. If $t_1,t_2,t_3$ be the tangents of the angle subtended by these parts at the opposite vertices. The relationship between $t_1,t_2,t_3$ is given by the following equation
$(\frac{1}{t_1}+\frac{1}{t_2})(\frac{1}{t_2}+\frac{1}{t_3})=k(1+\frac{1}{t_2^2})$. Then the value of $k$ is

Q. The tangent & Normal at any point P of the parabola intersect the axis at T $\&$ G. Centre of the circle circumscribing triangle PTG is the vertex of the parabola.

1. True

2. False

Q. Let $x^2+y^2-4x-2y-11=0$ be a circle. A pair of tangents from the point $(4,5)$ with a pair of radii form a quadrilateral of area sq. units.

Q.

A pair of tangents are drawn from a point on the directrix to a parabola $y^2=4ax$. The angle formed by the tangents will always be ${90}^{\circ }$

1. True

2. False

Q. If the tangents are drawn at $(a{t}_1^2,2a{t}_1)$ and $(a{t}_2^2,2a{t}_2)$ on the parabola $y^2=4ax$ intersect on axis of the parabola, then

1. $t_1{t}_2=2$
2. $t_1=-t_2$
3. $t_1{t}_2=-1$
4. $t_1{t}_2=-4$

Q. The equation to the pair of tangents drawn from (–1, –2) to parabola $x^2=2y$ is

1. 
2. 
3. 
4. 

Q.

If OA and OB are two equal chords of the circle $x^2+y^2-2x+4y=0$ perpendicular to each passing through the origin O, the slopes of OA and OB are the roots of the equation

1. 

2. 

3. 

4. 

Q. $PG$ is the normal at $P$ to the parabola $y^2=4ax$. $G$ is on the axis, $GP$ is produced to $Q$ such that $PQ=GP$, then

1. Locus of $Q$ is $y^2=16a(x+2a)$
2. Focus of locus of $Q$ is $(2a,0)$
3. Latus rectum of locus of $Q$ is $16a$
4. Locus of $Q$ is $y^2=16a(x-2a)$

Q. The tangents at the exterimities of any focal chord of a parabola intersect at right angle at the directrix.

1. True
2. False

Q. One of the sides of a triangle is divided into segments of $4$ and $6$ units by the point of tangency of the inscribed circle which has radius $2\sqrt{2}$ units, then the largest side of triangle is -

1. $10$
2. $\frac{21}{2}$
3. $\frac{43}{4}$
4. $11$

Q. Let a circle whose center on the axes touches the parabola $y^2=4x$ at two points such that pair of common tangents of the curves makes an angle of $\frac{\pi }{2}$. If the area of the circle is $k\pi$, then the value of $k$ is

Q. Find the shortest distance of the point $(0,c)$ from the parabola $y=x^2$, where $c>0$

Q. Let the tangents from $P$ are drawn to a circle with centre $C$ such that $PC=2$ units. If equation of the tangents are $x^2=3y^2,$ then radius of the circle can be

1. $\sqrt{2}$ unit
2. $\sqrt{3}$ unit
3. $2$ unit
4. $1$ unit