Double Ordinate

Q. Let $P$ be a variable point on the parabola $y=4x^2+1.$ Then, the locus of the mid-point of the point $P$ and the foot of the perpendicular drawn from the point $P$ to the line $y=x$ is:

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

Q. The chord of contact of tangents drawn from any point on the circle $x^2+y^2=a^2$ to the circle $x^2+y^2=b^2$ touches $x^2+y^2=c^2$, where $a,b,c>0,$ then $a,b,c$ are in

1. $b=\sqrt{ac}$
2. $2b=a+c$
3. $b=\frac{2ac}{a+c}$
4. $b^2=a^2+c^2$

Q. Length of chord of contact of $(2,5)$ with respect to $y^2=8x$ is

1. $\frac{3\sqrt{41}}{2}$ units
2. $\frac{2\sqrt{17}}{3}$ units
3. $\frac{7\sqrt{3}}{5}$ units
4. $\frac{5\sqrt{3}}{7}$ units

Q. Let a line $L:2x+y=k,k>0$ be a tangent to the hyperbola $x^2-y^2=3.$ If $L$ is also a tangent to the parabola $y^2=\alpha x,$ then $\alpha$ is equal to:

1. $-24$
2. $24$
3. $12$
4. $-12$

Q. If the focus of a parabola is $(2,5)$ and the directrix is $y=3$, the equation of the parabola is

1. $x^2+4x+4y+20=0$
2. $x^2+4x-4y+20=0$
3. $x^2-4x-4y+20=0$
4. $x^2+4x+4y+10=0$

Q. The circle $C_1:x^2+y^2=3$, with centre at $O$, intersect the parabola $x^2=2y$ at the point $P$ in the first quadrant. Let the tangent to the circle $C_1$ at $P$ touches other two circles $C_2$ and $C_3$ at $R_2$ and $R_3$, respectively. Suppose $C_2$ and $C_3$ have equal radii$\left(2\sqrt{3}\right)$ and centres $Q_2$ and $Q_3$ respectively. If $Q_2$ and $Q_3$ lie on the y-axis, then:

1. $Q_2{Q}_3=12$
2. $R_2{R}_3=4\sqrt{6}$
3. Area of the triangle $O{R}_2{R}_3$ is $6\sqrt{2}$
4. Area of the triangle $P{Q}_2{Q}_3$ is $4\sqrt{2}$

Q.

The locus of the points of trisection of the double ordinates of a parabola is a

1. pair of lines

2. circle

3. parabola

4. straight line

Q.

The points on the curve $y=2x^2-6x-4$ at which the tangent is parallel to the $x-$axis is

1. $\left(\frac{3}{2},\frac{13}{2}\right)$

2. $\left(-\frac{5}{2},-\frac{17}{2}\right)$

3. $\left(\frac{3}{2},\frac{17}{2}\right)$

4. $(0,–4)$

5. $\left(\frac{3}{2},-\frac{17}{2}\right)$

Q. The locus of the point of intersection of tangents drawn at the extremities of a normal chord to the parabola $y^2=4ax$ is the curve

1. $y^2(x+2a)-4a^3=0$
2. $y^2(x+2a)+4a^3=0$
3. $y^2(x-2a)+4a^3=0$
4. $y^2(x-2a)-4a^3=0$

Q. If the locus of the circumcentre of a variable triangle having sides $y-$axis, $y=2$ and $lx+my=1$, where $(l,m)$ lies on the parabola $y^2=4ax$ is a curve $C$, then the length of smallest focal chord of this curve $C$ (in units) is

1. $\frac{1}{4a}$
2. $\frac{1}{16a}$
3. $\frac{1}{12a}$
4. $\frac{1}{8a}$

Q. If $p$ be the perpendicular distance of a focal chord $PQ$ of length $l$ from the vertex $A$ of the parabola $y^2=4ax$, then $l$ varies inversely as

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

Q. The locus of the point of intersection of tangents drawn at the extremities of a normal chord to the parabola $y^2=4ax$ is the curve

1. $y^2(x+2a)+4a^3=0$
2. $y^2(x-2a)+4a^3=0$
3. $y^2(x+2a)-4a^3=0$
4. $y^2(x-2a)-4a^3=0$

Q. Let $y=f\left(x\right)$ be a parabola, having its axis parallel to $y-$axis, which is touched by the line $y=x$ at $x=1$, then

1. $f^{\prime }\left(0\right)=1-2f\left(0\right)$
2. $f\left(0\right)+f^{\prime }\left(0\right)+f^{\prime }\left(1\right)=1$
3. $f^{\prime }\left(1\right)=1$
4. $f^{\prime }\left(0\right)=f^{\prime }\left(1\right)$

Q. Tangent are drawn from $P$ to $y^2=4ax.$ If the difference of the ordinates of the point of contact of the two tangents is $l$, then the equation of the locus of $P$ is

1. $y^2-4ax=l^2$
2. $y^2-4ax=\frac{l^2}{2}$
3. $y^2-4ax=\frac{l^2}{4}$
4. $y^2-4ax=2l^2$

Q. The parabola $y^2=4ax$ passes through the centre of the circle $4x^2+4y^2-8x+12y-7=0$. The directrix of the parabola will be

1. $4x+9=0$
2. $4x+15=0$
3. $16x+9=0$
4. $16x-7=0$

Q.

For the parabola $y^2=4px$ find the extremities of a double ordinate of length 8p. Prove that the lines from the vertex to its extremities are at right angles.

Q. Length of the latus rectum of the parabola $y=4x^2-4x+3$ will be

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

Q. A tangent to the parabola $y^2+4x=0$ meets the parabola $y^2=8x$ in $P$ and $Q$. Then the locus of the middle point of $PQ$ is

1. Ellipse
2. Circle
3. Parabola
4. Hyperbola

Q. The locus of the vertices of the family of parabola $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$, $a$ being parameter is :

1. $xy=\frac{3}{4}$
2. $xy=\frac{105}{64}$
3. $xy=\frac{64}{105}$
4. $xy=\frac{35}{16}$

Q. If the locus of the circumcentre of of variable triangle having sides $y-$axis, $y=2$ and $lx+my=1$, where $(l,m)$ lies on the parabola $y^2=4ax$ is a curve $C$, then the curve $C$ is symmetric about the line

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

Q.

The coordinates $\left(5,2\right)$ and $\left(-6,2\right)$ are vertices of a hexagon.

Explain how to find the length of the segment formed by these endpoints.

How long is the segment?

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 $\tan \theta =$

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 locus of the point which divides the double ordinate of the parabola $y^2=6ax$ in the ratio $5:6,$ is

1. $100y^2=6ax$
2. $144y^2=6ax$
3. $121y^2=6ax$
4. $\frac{y^2}{121}=6ax$

Q. The locus of the vertices of the family of parabolas $y=\frac{a^3{x}^2}{3}+\frac{a^{2}x}{2}-2a$ is

1. 
2. 
3. 
4. 

Q. The common tangent between the curves $x^2+y^2=1$ and $y^2=4(x-2)$, passing through $(x_1,y_1)$ on $y^2=4(x-2)$ satisfies $a{x}_1^2-9x_1+b=0$. Then $a+b$ is equal to

Q. Let $ABC$ be an acute scalene triangle, and $O$ and $H$ be its circumcentre and orthocentre respectively. Further let $N$ be the midpoint of $OH$. The value of the vector sum $\overrightarrow{NA}+\overrightarrow{NB}+\overrightarrow{NC}$ is

1. $\overrightarrow{HO}$
2. $\overrightarrow{0}$ (zero vector)
3. $\frac{1}{2}\overrightarrow{HO}$
4. $\frac{1}{2}\overrightarrow{OH}$

Q. Find the equation of parabola whose
Focus is $(3,0)$ and the directrix is $3x+4y=1$

Q. The equation of the parabola whose focus is $(-3,0)$ and the directrix is $x+5=0$ is :

1. $y^2=4\left(x-4\right)$
2. $y^2=2\left(x+4\right)$
3. $y^2=4\left(x-3\right)$
4. $y^2=4\left(x+4\right)$

Q. For parabola $x^2+y^2+2xy-6x-2y+3=0$, the focus is

1. $(1,-1)$
2. $(-1,1)$
3. $(3,1)$
4. $noneofthese$

Q.

The equation of the parabola with focus (3, 0) and the directirx x +x3 = 0 is

1. 

2. 

3. 

4.