@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.