Given, curves are A:4x2+9y2=36 B:(2x)2+(2y)2=31 L is common tangent line to curve A and B. Let slope of L be m. Given, curve A is equation of ellipse.
x2
9
+
y2
4
=1 Then, equation of tangent to ellipse will be y=mx±√9m2+4. . . (i) Curve B is circle i.e x2+y2=
31
4
Then, equation of tangent to circle will be y=mx±√(
31
4
)m2+(
31
4
) . . . (ii) On comparing Eqs. (i) and (ii), we get On comparing Eqs. (i) and (ii), we get 9m2+4=(
31
4
)m2+(
31
4
) ⇒36m2+16=31m2+31⇒5m2=15 ⇒m2=3 ∴ Square of slope of line L is 3 .
