$\text{ Tangent is drawn at any point }\left(x_1,y_1\right)\text{ other than vertex on the }$$\text{ parabola }{y}^2=4cx\text{ . If tangents are drawn from any point on this tangent to the }$$\text{ circle }{x}^2+y^2=a^2\text{ such that all the chords of contact pass through a fixed point }$$\left(x_2,y_2\right)\text{ then }$

$\begin{array}{l}\text{ Let }\left(x_1,y_1\right)=\left(a{t}^2,2at\right)\\ \text{ Tangent at this point is }ty=x+a{t}^2\text{. }\end{array}$

$\text{ Any point on this tangent is }\left(h,\frac{h+a{t}^2}{t}\right)\text{. }$

$\text{ Chord of contact of this point with respect to the circle }{x}^2+y^2=a^2\text{ is: }$

$hx+\left(\frac{h+a{t}^2}{t}\right)y=a^2\text{ or }\left(\text{ at }y-a^2\right)+h\left(x+\frac{y}{t}\right)=0$

which is a family of straight lines passing through point of intersection of 

$\begin{array}{l}ty-a=0\text{ and }x+\frac{y}{t}=0\text{. }\\ \text{ So, the fixed point is }\left(-\frac{a}{t^2},\frac{a}{t}\right)\\ \therefore x_2=-\frac{a}{t^2},y_2=\frac{a}{t}.\text{ Clearly, }{x}_1{x}_2=-a^2,y_1{y}_2=2a^2.\end{array}$