Chord with a Given Mid Point : Hyperbola

Q. If the chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax,$ then the locus of the midpoints of the chords is the curve

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

Q. The locus of mid point of the chords of $x^2-y^2=4,$ that also touches the parabola $y^2=8x$ is

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

Q. A pair of variable straight lines $5x^2+3y^2+\alpha xy=0(\alpha \in R)$, cut the parabola $y^2=4x$ at two points (other than origin) $P$ and $Q$. If the locus of the point of intersection of tangents to the given parabola at $P$ and $Q$ is given by $x=\frac{p}{q}$ (where $p,q$ are natural numbers coprime to each other), then the value of $(p–q+2)$ is equal to

1. 6
2. 9
3. 4
4. 5

Q. If a variable chord of $x^2–y^2=9$ touches $y^2=12x$ and locus of middle point of these chords is $x^3+{\lambda }_{1}x{y}^2+{\lambda }_2{y}^2=0$, then the value of ${\lambda }_2-{\lambda }_1$ is

Q. Equation of chord of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose mid point is $(x_1,y_1)$ is given by

1. $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}-1=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1$
2. $\frac{x_1{x}^2}{a^2}-\frac{y_1{y}^2}{b^2}-1=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1$
3. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}$
4. $\frac{x+x_1}{a^2}-\frac{(y+y_1)}{b^2}=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}$

Q. The eccentricity of the conic $4(2y-x-3{)}^2-9(4x+2y-1{)}^2=80$ is

1. $\frac{5}{3}$
2. $\frac{4}{3}$
3. $\frac{\sqrt{11}}{3}$
4. $\frac{\sqrt{10}}{3}$

Q. Let $S=x^2+y^2+2gx++2fy+c=0$ be a given circle. Find the locus of the foot of the perpendicular drawn from origin upon any chord which subtends a right angle at the origin.

1. $2(x^2+y^2)+2(9x+fy)+c=0$
2. $x^2+y^2+2(9x+fy)+c=0$
3. $2x^2+2y^2+9xfy+c=0$
4. None of these

Q. Equation of chord of the hyperbola $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ whose mid point is $(x_1,y_1)$ is given by

1. $\frac{x{x}_1}{a^2}-\frac{y{y}_1}{b^2}-1=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1$
2. $\frac{x_1{x}^2}{a^2}-\frac{y_1{y}^2}{b^2}-1=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}-1$
3. $\frac{x^2}{a^2}-\frac{y^2}{b^2}=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}$
4. $\frac{x+x_1}{a^2}-\frac{(y+y_1)}{b^2}=\frac{x_1^2}{a^2}-\frac{y_1^2}{b^2}$

Q. If a variable chord of $x^2–y^2=9$ touches $y^2=12x$ and locus of middle point of these chords is $x^3+{\lambda }_{1}x{y}^2+{\lambda }_2{y}^2=0$, then the value of ${\lambda }_2-{\lambda }_1$ is

Q. If a variable line has its intercepts on the coordinates axes as $e,e^{\prime }$ where $\frac{e}{2},\frac{e^{\prime }}{2}$ are the eccentricities of a hyperbola and its conjugate hyperbola respectively, then the line always touches the circle centred at $O$ whose radius $r$ is equal to

Q. If the chords of the hyperbola $x^2-y^2=a^2$ touch the parabola $y^2=4ax,$ then the locus of the midpoints of the chords is the curve

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