Focal Chord

Q.

The equation of the common tangent to the curves $y^2=8x$ and $xy=-1$is

1. $3y=9x+2$

2. $y=2x+1$

3. $2y=x+8$

4. $y=x+2$

Q. Let normals drawn to parabola at points $P(0,0)$ and $Q(3,-1)$ intersect at $(2,1)$. If $PQ$ is bisected by the axis of the parabola, then

1. Equation of directrix is $x+3y+5=0$
2. Slope of axis is $3$
3. Focus is $(8,0)$
4. Slope of tangent at vertex is $\frac{1}{3}$

Q. Let $PQ$ be a focal chord of $y^2=4ax$. The tangents to parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a(a>0)$. If the chord $PQ$ subtends an angle $\theta$ at the vertex of prabola then $\tan \theta =$

1. $\frac{2\sqrt{7}}{3}$
2. $\frac{2\sqrt{7}}{5}$
3. $\frac{2\sqrt{5}}{3}$
4. $\frac{2\sqrt{3}}{5}$

Q. Let $PQ$ be a focal chord of parabola $y^2=x.$ If the coordinates of $P$ is $(4,-2),$ then the slope of the tangent at $Q$ is

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

Q. If tangents are drawn to the parabola $(x-3{)}^2+(y+4{)}^2=\frac{(3x-4y-6{)}^2}{25}$ at the extremities of the chord $2x-3y-18=0$, then angle between tangents is

1. ${45}^{\circ }$
2. ${90}^{\circ }$
3. ${60}^{\circ }$
4. ${120}^{\circ }$

Q. $(\frac{1}{2},2)$ is one extremity of a focal chord of the parabola $y^2=8x$. The coordinates of the other extremity is

1. (8, 8)
2. (–8, 8)
3. (8, –8)
4. (–8, –8)

Q. Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,a>0$.
If chord $PQ$ subtends an angle $\theta$ at the vertex of $y^2=4ax$, then tan $\theta$ is equal to

1. $\frac{2}{3}\sqrt{7}$
2. $\frac{-2}{3}\sqrt{7}$
3. $\frac{2}{3}\sqrt{5}$
4. $\frac{-2}{3}\sqrt{5}$

Q. Consider $P$ is a point on $y^2=4ax,$ if the normal at $P,$ the axis and the focal radius of $P$ form an equilateral triangle. Then coordinates of $P$ are

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

Q. The lie $x-b+\lambda y=0$ cuts the parabola $y^2=4ax$ at $P\left(t_1\right)$ and $Q\left(t_2\right)$. If $b\in [2a,4a]$ and $\lambda \in R$, then the value(s) $\left[t_1{t}_2\right]$ is/are
(where $[.]$ repreasents the greatest integer function and $t_1,t_2$ are parametic points)

1. $-5$
2. $-4$
3. $-3$
4. $-2$

Q. Let $PQ$ be a focal chord of the parabola $y^2=4ax$. The tangents to the parabola at $P$ and $Q$ meet at a point lying on the line $y=2x+a,a>0$.
Length of chord $PQ$

1. 7a
2. 5a
3. 2a
4. 3a

Q. For the rectangular hyperbola $xy=25$

1. Conjugate axis
2. $y=-x$
3. Transverse axis
4. $(5,5)(-5,-5)$
5. Vertices
6. Centre
7. $(0,0)$
8. $y=x$

Q. The focal chord to $y^2=16x$ is tangent to $(x-6{)}^2+y^2=2$, then the possible values of the slope of this chord are

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

Q. If for parabola $y^2=4ax$ half of the length of focal chord is $8a$, then the angle made by focal chord with $x-$ axis is/are

1. $\frac{\pi }{6}$
2. $\frac{3\pi }{4}$
3. $\frac{5\pi }{6}$
4. $\frac{2\pi }{3}$

Q. The length of a focal chord of the parabola $y^2=4ax$ at a distance $b$ from the vertex is $c$, then

1. $2a^2=bc$
2. $a^3=b^{2}c$
3. $ac=b^2$
4. $b^{2}c=4a^3$

Q. If slope of the focal chord of parabola $y^2=8x$ is the greater root of the equation $x^2-6x+8=0$, then length of the focal chord will be

1. $10$ units
2. $17$ units
3. $\frac{17}{4}$ units
4. $\frac{17}{2}$ units

Q. Let $a,r,s,t$ be nonzero real numbers. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)\text{and}S(a{s}^2,2as)$ be distinct points on the parabola $y^2=4ax$. Suppose that $PQ$ is the focal chord and lines $QR$ and $PK$ are parallel, where $K$ is the point $(2a,0).$

The value of $r$ is

1. $-\frac{1}{t}$
2. $\frac{t^2+1}{t}$
3. $\frac{1}{t}$
4. $\frac{t^2-1}{t}$

Q. The locus of the point of intersection of the tangents to the parabola $y^2=4ax$ which makes angles ${\theta }_1$ and ${\theta }_2$ with its axis so that $\cot {\theta }_1+\cot {\theta }_2=k$ is

1. $kx-y=0$
2. $kx-a=0$
3. $y-ka=0$
4. $x-ka=0$

Q. If tangents are drawn to the parabola $(x-2{)}^2+(y-3{)}^2=\frac{(3x+4y-5{)}^2}{25}$ at the extremities of the chord $3x-y-3=0$, then angle between the tangents is

1. ${45}^{\circ }$
2. ${90}^{\circ }$
3. ${60}^{\circ }$
4. ${120}^{\circ }$

Q. Let $A$ be the vertex of the parabola $y^2=4ax$ and the length of focal chord $PQ$ is $l\text{units}$. If perpendicular distance from vertex to $PQ$ is $p\text{units}$, then

1. $l$ is proportional to $p^2$
2. $l$ is inversely proportional to $p^2$
3. $l$ is proportional to $p$
4. $l$ is inversely proportional to $p$