Focal Chord

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

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.

PSQ is a focal chord of the parabola $y^2=8x$.If SP=6, then write SQ.

Q. If $(x_1,y_1)$ and $(x_2,y_2)$ are the extremeties of the focal chord of parabola $y^2=16ax$, then the value of $16x_1{x}_2+y_1{y}_2$ is

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. Let normal's drawn to parabola at point's $P(0,0)$ 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 $a,r,s$ and $t$ be non –zero real numbers. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)$ 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 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. 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. 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. $(\frac{1}{2},2)$ 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. 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. $y_1+y_2-y_3-y_4=0$

Q. Let $P(a{t}^2,2at),Q,R(a{r}^2,2ar)$ be three points on a parabola $y^2=4ax.$ If $PQ$ is the focal chord and $PK,QR$ are parallel where the co-ordinates of $K$ is $(2a,0),$ then the value of $r$ is

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

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

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. ${120}^{\circ }$
4. ${60}^{\circ }$

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. The point $(1,2)$ is one extremity of focal chord of the parabola $y^2=4x$. The length of this focal chord is

1. $1$
2. $2$
3. $4$
4. $8$

Q. If $b,k$ are the intercepts of the focal chord of $y^2=4ax$, $a\ne b$ then $k$ is

1. $\frac{ab}{a-b}$
2. $\frac{2ab}{b-a}$
3. $\frac{ab}{b-a}$
4. $\frac{2ab}{a-b}$

Q. The equation of the common tangent to the curve $y^2=4x$ and $xy=16$ is

1. $x+4y+16=0$
2. $x+8y+64=0$
3. $4x+y+16=0$
4. $8x+y+64=0$

Q.

The sum of the reciprocals of the focal distance of a focal chord $PQ$ of $y^2=4ax$ is

1. $a$

2. $\frac{1}{a}$

3. $2a$

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

Q.

Prove that the curves xy = 4 and $x^2+y^2=8$ touch each other.

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. 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. 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 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. If a circle whose one end of the diameter is focus of the parabola $y^2=4x$ and other end is a point on parabola, then which of the following line will always be a tangent to the given circle?

1. $x=0$
2. $x=-1$
3. $x=1$
4. $y=0$

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 lines $\sqrt{3}x-y-4\sqrt{3}\lambda =0$ and $\sqrt{3}\lambda x+\lambda y-4\sqrt{3}=0$ is a hyperbola of eccentricity

Q. A square with side length $1\text{cm}$ has an inscribed semicircle along one of its sides as shown in figure. The line $AE$ is tangent to the semicircle. What is the length of $AE$ ?

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

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 $y=2x+a(a>0)$. Length of $PQ$(in units) is :

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

Q. From a point P(1, 2), two tangents are drawn to a hyperbola H in which one tangent is drawn to each arm 0f the hyperbola. If the equations of asymptotes of hyperbola H are $\sqrt{3}x-y+5=0and\sqrt{3}x+y-1=0$, then eccentricity of H is

1. 
   
2. 
3. 
   
4. 2