A normal to the parabola  y2=4ax with slope m touches the rectangular hyperbola x2−y2=a2 .  If

Equation of the normal to the hyperbola y2=4ax  with a slope m is y=mx−2am−am3 . It touches  the rectangular hyperbola ,x2−y2=a2    if   (−2am−am3)=a2(m2−1) ⇒a2m2(2+m2)2=a2(m2−1),  ⇒m2[m4+4m2+4]=m2−1 m6+4m4+4m2−m2+1=0 ⇒m6+4m4+3m2+1=0