Length of Chord

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. 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. $y^2=4x$ 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.

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. 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. Consider a circle $S:{(x+2)}^2+{(y-8)}^2=16$ and a parabola $P:y^2=8x.TA$ and $TB$ are two tangents drawn from a point $T$ on the parabola $P=0$ to the circle $S=0$ such that $TA+TB$ is minimum. $A$ and $B$ are the points of contact.

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}T\equiv (a,b),\text{then}(a+b)\text{equals}& \text{(P)}& 6\\ \text{(B)}& \text{Sum of ordinates of}A\text{and}B\text{is}& \text{(Q)}& 8\\ \text{(C)}& \text{Minimum value of}(TA+TB)\text{is}& \text{(R)}& 10\\ \text{(D)}& \text{Area of}\Delta TAB\text{is}& \text{(S)}& 12\end{array}$

Which of the following is a CORRECT combination ?

1. $\text{(C)}\to \text{(S)},\text{(D)}\to \text{(R)}$
2. $\text{(C)}\to \text{(R)},\text{(D)}\to \text{(S)}$
3. $\text{(C)}\to \text{(P)},\text{(D)}\to \text{(Q)}$
4. $\text{(C)}\to \text{(Q)},\text{(D)}\to \text{(Q)}$

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 $A$ and $B$ be two distinct points on the parabola $y^2=4x$. If the axis of the parabola touches a circle of radius $r$ having $AB$ as its diameter, then the slope of the line joining $A$ and $B$ can be

1. $\frac{2}{r}$
2. $-\frac{2}{r}$
3. $\frac{1}{r}$
4. $-\frac{1}{r}$

Q. If the line $y=mx+c$ is a common tangent to the hyperbola $\frac{x^2}{100}-\frac{y^2}{64}=1$ and the circle $x^2+y^2=36$, then which one of the following is true?

1. $4c^2=369$
2. $c^2=369$
3. $8m+5=0$
4. $5m=4$

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.

Let $L_1$ be a straight line passing through the origin and $L_2$ be the straight line x + y=1. If the intercepts made by the circles $x^2+y^2-x+3y=0$ on $L_1$ and $L_2$ are equal then which of the following equations can represent $L_1?$

1. x+y=0
2. x-7y=0
3. 7x-y=0
4. x-y=0

Q. The numbers of $3\times 3$ matrices A whose entries are either $0$ or $1$ and for which the system $A\left[\begin{array}{c}x\\ y\\ z\end{array}\right]=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$ has exactly two distinct solutions is?

1. $0$
2. $2^9-1$
3. $2$
4. $168$

Q. If $P(-3,2)$ is one end of the focal chord $PQ$ of the parabola $y^2+4x+4y=0$, then the slope of the normal at $Q$ is

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

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 two normals to a parabola $y^2=4ax$ intersect at right angles, then the chord joining their feet passes through a fixed point whose co-ordinates are

1. $(2a,0)$
2. $(-2a,0)$
3. $(a,0)$
4. none of these

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

Q. The area bounded by the tangent on the curve
$y=4x^2+2x\text{at (0, 0)},y-10=-x\text{and}y=0$ is

1. $\frac{100}{9}\text{sq.units}$
2. $20\text{sq.units}$
3. $\frac{100}{3}\text{sq.units}$
4. $\frac{200}{9}\text{sq.units}$

Q. Any chord $ABofy62=4ax$ cuts the axis of the parabola at $P$, so that $AP:AB=1:3whereA(a{t}_1^2,2a{t}_1),B(a{t}_2^2,2a{t}_2)$, then

1. $2t_1+t_2=0$
2. $t_1+2t_2=0$
3. $t_1{t}_2=-4$
4. $t_1+t_2=0$

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

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. Consider a circle $S:{(x+2)}^2+{(y-8)}^2=16$ and a parabola $P:y^2=8x.TA$ and $TB$ are two tangents drawn from a point $T$ on the parabola $P=0$ to the circle $S=0$ such that $TA+TB$ is minimum. $A$ and $B$ are the points of contact.

$\begin{array}{cccc}& \text{List I}& & \text{List II}\\ \text{(A)}& \text{If}T\equiv (a,b),\text{then}(a+b)\text{equals}& \text{(P)}& 6\\ \text{(B)}& \text{Sum of ordinates of}A\text{and}B\text{is}& \text{(Q)}& 8\\ \text{(C)}& \text{Minimum value of}(TA+TB)\text{is}& \text{(R)}& 10\\ \text{(D)}& \text{Area of}\Delta TAB\text{is}& \text{(S)}& 12\end{array}$

Which of the following is a CORRECT combination ?

1. $\text{(A)}\to \text{(S)},\text{(B)}\to \text{(R)}$
2. $\text{(A)}\to \text{(P)},\text{(B)}\to \text{(S)}$
3. $\text{(A)}\to \text{(S)},\text{(B)}\to \text{(P)}$
4. $\text{(A)}\to \text{(Q)},\text{(B)}\to \text{(R)}$

Q. If a focal chord of the parabola $y^2=ax$ is $2x-y-8=0$, then the equation of the directrix is

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