The equation of the normal to the hyperbola x24−y21=1 at (2secθ,tanθ) is 2xcosθ+ycotθ=5 ...(i)
Normal has equal intercepts on positive x−and y−axes.
Thus slope of the normal is −2sinθ=−1
⇒sinθ=12 ⇒θ=π6,5π6
But θ≠5π6 as normal lies in Q1
From (i), Equation of line will be y=−x+5√3
As it touches the ellipse x2a2+y2b2=1,
we have c2=a2m2+b2
Here c=5√3, m=−1
⇒253=a2(−1)2+b2
⇒253=a2+b2
⇒3(a2+b2)=25