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不论是用几何方法,还是用代数方法,抑或是用数形结合方法,都不能忽视“数”的存在,也不能忽视“形”的存在;只凭主观想象,无中生有,则会导致错解.例1已知抛物线y2=x有一条长为t的动弦AB,求AB的中点M到y轴的最短距离dmin.错解:如图1,F(1/4,0)为抛物线的焦点,l为其准线,AE⊥l,BG⊥l,E,G为垂足,则xM+1/4=|MN|=1/2(|AE|+|BG|)=1/2(|AF|+|BF|)≥1/2|AB|=t/2.(*)∴xM≤t/2-1/4,即dmin=t/2-1/4.
Whether it is a geometric method, an algebraic method, or a combination of number and shape methods, we cannot ignore the existence of “numbers” and ignore the existence of “forms.” Only subjective imaginations, out of nothing, can lead to misunderstandings. Example 1 It is known that the parabola y2=x has a chord AB of length t. Find the shortest distance dmin between the middle point of AB and the y-axis. Mistake: As shown in Figure 1, F(1/4,0) is a parabola. Focus, l is its guideline, AE⊥l, BG⊥l, E, G is a pedicle, then xM+1/4=|MN|=1/2(|AE|+|BG|)=1/2 (|AF|+|BF|)≥1/2|AB|=t/2.(*)∴xM≤t/2-1/4, that is, dmin=t/2-1/4.