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. 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