猫猫的猪宝宝
春芽
共回答了21个问题采纳率:85.7% 举报
(1)设OP交AB于Q,易知OP*OQ=OA^2,OP垂直平分AB,
∴OQ/OP=b^2/OP^2,
∴Q(b^2*x0/(x0^2+y0^2),b^2*y0/(x0^2+y0^2)),
AB的斜率=-x0/y0,
∴AB的方程为x0x+y0y=b^2.
(2)椭圆的短轴长为8,∴b=4,
∴直线AB交X轴于M(16/x0,0),交Y轴于N(0,16/y0),
由a^2/OM^2+b^2/ON^2=25/16得
a^2*x0^2+16y0^2=400,(1)
又y0^2/a^2+x0^2/16=1.(2)
(1)-(2)*400,得
(a^2-25)(x0^2+16y0^2/a^2)=0,x0y0≠0,
∴a^2=25,
∴椭圆C的方程为y^2/25+x^2/16=1.
(3)PA⊥PB,
OP^2=2OA^2=2b^2,
{x^2+y^2=2b^2,y^2/a^2+x^2/b^2=1},
{y^2=2b^2-x^2,x^2=b^2(a^2-2b^2)/(a^2-b^2)>0},
a>b>0,
==>a>b√2.
1年前
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