Double Ordinate

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. Find the equation of the chord joining the points $P(\frac{\pi }{4})\text{and}Q(\frac{\pi }{4})$ on the ellipse $\frac{x^2}{25}+\frac{y^2}{9}=1.$

1. $3x+5y=15\sqrt{2}$
2. $3x-5y=15\sqrt{2}$
3. $5x+3y=15\sqrt{2}$
4. $5x-3y=15\sqrt{2}$

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. 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. $\frac{1}{12a}$
2. $\frac{1}{4a}$
3. $\frac{1}{16a}$
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. 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. The equation of tangent to the hyperbola
$xy=16$ at the point $\text{P}\left(2\right)$ in the first quadrant is:

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

Q.

An equilateral triangle is inscribed in the parabola $y^2=4ax,$ such that one vertex of this triangle coincides with the vertex of the parabola. Side length of this triangle is

1. $4a\sqrt{3}$

2. $6a\sqrt{3}$

3. $2a\sqrt{3}$

4. $8a\sqrt{3}$

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