设抛物线y=2px(p>0)的焦点为F,O为准线上一点 1若tan∠OQF=1/3,求Q点坐标 2求cos∠OQF的最大
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P>0 F(0.5p,0) 准线x=-0.5p与X轴交于点A(-0.5p,0),设Q(-0.5p,y),则 AF=p,AO=0.5p ∠OQF=∠AOQ-∠AFQ tan∠AOQ=|y/(0.5p)|=|2y/p|,tan∠AFQ=|y/p| tan∠OQF=tan(∠AOQ-∠AFQ)=(|2y/p|-|y/p|)/(1+|2y/p|*|y/p|)=1/3 p|y|/(p^2+2y^2)=1/3 y=0.5p,p,-0.5p,-p Q点坐标有4个:(-0.5p,0.5p),(-0.5p,p),(-0.5p,-0.5p),(-0.5p,-p) 要∠OQF的最大值,则p|y|/(p^2+2y^2)=最大值 设p|y|/(p^2+2y^2)=s,则 2sy^2-p|y|+sp^2=0 未知数y有实数解,则它的判别式/△≥0,即 (-p)^2-4*2s*sp^2≥0 s^2≤1/8 s最大=1/√8 ∠OQF的最大值=arctan(1/√8)
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