Focal Chord

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. If one end point of the focal chord of the parabola $y^2=4ax$ is $(1,2)$, then the other end point lies on

1. $x^{2}y+2=0$
2. $xy+2=0$
3. $xy-2=0$
4. $x^2+xy-y-1=0$

Q. If $PSQ$ is the focal chord of the parabola $y^2=8x$ such that $SP=6$. Then the length of $SQ$ is

1. $6$
2. $4$
3. $3$
4. None of these

Q. If the point $P(4,-2)$ is one end of the focal chord $PQ$ of the parabola $y^2=x$, then the slope of the tangent at $Q$ is

Q. Equation of the parabola obtained by taking reflection of $y=4x^2-4x+3$ about the line $y=x,$ will be

1. ${(y-\frac{1}{2})}^2=\frac{1}{4}(x-2)$
2. ${(y-\frac{1}{2})}^2=4(x-2)$
3. ${(y-2)}^2=\frac{1}{4}(x-\frac{1}{2})$
4. ${(y-2)}^2=4(x-\frac{1}{2})$

Q. A line $L$ passing through the focus of the parabola $y^2=4(x-1)$, intersects the parabola in two distinct points. If ${}^{\prime }{m}^{\prime }$ be the slope of the line ${}^{\prime }{L}^{\prime }$ then

1. $-1<m<1$
2. $m<-1$ or $m>1$
3. $m\in R$
4. None of the above

Q. The equation of common tangent to curve $xy=-1$ and $y^2=8x$ is

1. $3y=9x+2$
2. $y=2x+1$
3. $2y=x+8$
4. $y=x+2$

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 $y=2x+a,a>0$.
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. 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. The length of the intercept on the normal at the point $(a{t}^2,2at)$ of the parabola $y^2=4ax$ made by the circle which is described on the focal distance of the given point as diameter is

1. $a(1+t^2)$
2. $a\sqrt{1+t^2}$
3. none of these
4. $\sqrt{a(1+t^2)}$

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 $(x_r,y_r):r=1,2,3,4$ be the points of intersection of the parabola $y^2=4ax$ and the circle $x^2+y^2+2gx+2fy+c=0,$ then

1. $y_1+y_2+y_3+y_4=0$
2. $\sqrt{x_1}+\sqrt{x_2}+\sqrt{x_3}+\sqrt{x_4}=0$
3. $y_1-y_2+y_3-y_4=0$
4. $y_1+y_2-y_3-y_4=0$