Length of Chord

Q. $y^2=4x$ and $y^2=-8(x-a)$ intersect at point $A$ and $C$. Points $O(0,0),A,B(a,0),C$ are concyclic.
Tangents to parabola $y^2=4x$ at $A$ and $C$ intersect at point $D$ and tangents to parabola $y^2=-8(x-a)$ intersect at point $E$, then the area of quadrilateral $DAEC$ is

1. $48\sqrt{3}$
2. $96\sqrt{2}$
3. $36\sqrt{6}$
4. $54\sqrt{5}$

Q. Consider the parabola whose focus at $(0,0)$ and tangent at vertex is $x-y+1=0$.
The length of chord of a parabola on the $x-$axis is

1. $4\sqrt{2}$
2. $2\sqrt{2}$
3. $8\sqrt{2}$
4. $3\sqrt{2}$

Q. If the $x$-intercept of the focal chord of the parabola $y^2-2y-4x+5=0$ is $\frac{11}{4}$, then find the length of chord.

1. $\frac{15}{2}$
2. $\frac{25}{2}$
3. $\frac{25}{4}$
4. $\frac{15}{4}$

Q. If the length of the chord of the parabola $y^2=4x$ whose slope is $1$, is $10\sqrt{2}$ units, then equation of the chord is

1. $x=y+21$
2. $4x=4y+21$
3. $x=y-21$
4. $4y=4x+21$

Q. Let $P\left(t_1\right),Q\left(t_2\right)$ be two points on the parabola $y^2=8x.$ Then which of the following is/are true?

1. If the chord $PQ$ meets the $x-$axis at $(4,0)$, then $t_1{t}_2=-2$
2. If $PQ$ is a focal chord, then $t_1{t}_2=-2$
3. If $t_1=2$, then the length of focal chord $PQ$ is $\frac{25}{4}$
4. If $t_1=-\frac{1}{2}$ and $PQ$ meets the $x-$axis at $(4,0)$, then $Q\equiv (32,16)$

Q.

Write the length of the chord of the parabola $y^2=4ax$ which passes through the vertex and is inclined to the axis $\frac{\pi }{4}$.

Q.

In the parabola $y^2=4ax$, the length of the chord passing through the vertex and inclined to the axis at $\frac{\pi }{4}$ is

1. $4\sqrt{2}a$

2. $2\sqrt{2}a$

3. none of these

4. $\sqrt{2}a$

Q.

A chord 8 cm long is 3 cm away from the centre of the circle. What is the length of a chord which is 4 cm away from the centre?

1. 2 cm

2. 6 cm

3. 8 cm

4. 10 cm

Q.

Two parallel chords 10 centimeters and 24 centimeters long are drawn on the same side of the centre of a circle of radius 13 centimeters. Find the distance between the chords.

1. 5 cm

2. 6.5 cm

3. 7 cm

4. 8.5 cm

Q.

In the semicircle shown, the top chord is parallel to the diameter. What is its length?

1. 5 cm

2. 10 cm

3. 16 cm

4. 20 cm