Locus

Q. If one axis of varying standard hyperbola be fixed in magnitude and position, then the locus of the point of contact of tangents drawn to it from a fixed point $(0,c)$ on the other axis is

1. $y^2=\frac{a^2}{c}(x-c)$
   
2. $x^2=-\frac{a^2}{c}(y-c)$
3. $x^2=\frac{a^2}{c}(y+c)$
   
4. $y^2=-\frac{a^2}{c}(x-c)$
   

Q. A variable circle passes through the point $P(1,2)$ and touches the $x-$axis. The locus of the other end of the diameter through $P$ is

1. $(x-1{)}^2=8y$
2. $(x+1{)}^2=8y$
3. $(y-1{)}^2=8x$
4. $(x-1{)}^2+8y=0$

Q. If one axis of varying standard hyperbola be fixed in magnitude and position, then the locus of the point of contact of tangents drawn to it from a fixed point $(0,c)$ on the other axis is

1. $y^2=\frac{a^2}{c}(x-c)$
   
2. $x^2=-\frac{a^2}{c}(y-c)$
3. $x^2=\frac{a^2}{c}(y+c)$
   
4. $y^2=-\frac{a^2}{c}(x-c)$