The correct option is B Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.
The equation of ellipse is,
2x2+y2=4
∴2x24+y24=1
⇒x22+y24=1
comparing with standard form,
x2a2+y2b2=1
we get,
a2=2 and b2=4
and equation of tangent to ellipse x2a2+y2b2=1 is,
y=mx±√a2m2+b2
∴ y=mx±√2m2+4 ...(1)
also given that,
equation of tangent to the parabola y2=16√3x is,
y=mx+4√3m ...(2)
on comparing (1) and (2) we get,
4√3m=√2m2+4
4√3=m√2m2+4
squaring on both the sides,
48=2m4+4m2
∴2m4+4m2−48=0
⇒m4+2m2−24=0
m4+2m2=24 this equation satisfies statement-2
On solving this equation
m4+2m2−24=0
m4+6m2−4m2−24=0
m2(m2+6)−4(m2+6)=0
∴ we get,
m2+6=0 and m2−4=0
m2−4=0 ⇒m=±2
∴ the equation of common tangent are y=±2x±2√3
Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for statement-1.