Pair of Tangents from an External Point

Q.

If two tangents drawn from a point$P$ to the parabola $y^2=4x$ are at right angles, then the locus of$P$ is

1. $2x+1=0$

2. $x=-1$

3. $2x-1=0$

4. $x=1$

Q.

The equation of the tangent to the curve $\left(\frac{x^2}{3}\right)-\left(\frac{y^2}{2}\right)=1$ which is parallel to $y=x$, is

1. $y=x\pm 1$

2. $y=x-\left(\frac{1}{2}\right)$

3. $y=x+\left(\frac{1}{2}\right)$

4. $y=1-x$

Q.

The point of intersection of the tangents drawn to the curve $x^{2}y=1-y$ at the point where it is met by the curve $xy=1-y$ is given by

1. $\left(2,-1\right)$

2. $\left(1,2\right)$

3. $\left(0,1\right)$

4. None of these

Q. Let the equation $x^2-y^2-2x+2y=0$ represent the pair of tangents to the circle $2x^2-4x+2y^2+1=0$. If $a,L$ represent the lengths of tangents and equation of common chord respectively, then

1. $L:2y=1$
2. $a=1$ units
3. $a=\frac{1}{\sqrt{2}}$ units
4. $L:y=1$

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. Two tangents are drawn from the point $(-2,-1)$ to the parabola $y^2=4x$. If $\alpha$ is the angle between those tangents then tan $\alpha$=

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

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 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 the pair of straight lines $\sqrt{3}xy-x^2=0$ is tangent to the circle at $P$ and $Q$ from origin $O$ such that area of the smaller sector formed by $CP$ and $CQ$ is $3\pi$ sq. unit, where $C$ is the centre of circle, then $OP$ equals to

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

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. Let the equation $x^2-y^2-2x+2y=0$ represent the pair of tangents to the circle $2x^2-4x+2y^2+1=0$. If $a,L$ represent the lengths of tangents and equation of common chord respectively, then

1. $L:2y=1$
2. $a=1$ units
3. $a=\frac{1}{\sqrt{2}}$ units
4. $L:y=1$

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

1. True
2. False

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

1. $x^2-2xy-3y^2+10x+6y+9=0$
2. $4x^2-2xy-y^2+4x-6y-4=0$
3. $x^2+2xy+3y^2+10x+6y+9=0$
4. $4x^2+2xy-3y^2+4x-6y-4=0$

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