the point on the curve y = x^2  which is closest to the line 2x - y - 4 = 0

The point closest to the line is the point on the curve where the tangent is parallel to the given line.So,any tangent to the curve is 2xx₀=y+y₀.This gives x₀=1 and hence the point is (1,1)

Another method:distance=|2x-x²-4|/  differentiate =>x=1,y=1

@allamraju:why is it so..there may be a tangent that intersects the line too?this can't be a general statement isn't it

ans is(1,1)

@risin In fact,Many tangents intersect the line.But their point of contacts are definitely farther from the line than (1,1) and hence,we got the ans as (1,1).About the statement,I read it in some book and applied it many times for other curves too in many questions.If u wish to check,Plot a rough for the given question and observe,though many tangents intersect the line,their point of contacts with the parabola are at greater distance.Hope It is clear to u.