Question

椭圆焦点在x轴上 长轴为短轴的√3倍,直线y=x与椭圆交于A,B两点,C为椭圆的右顶点,向量OA与向量OC的内积=3/2

Answer 1

椭圆x^2/a^2+y^2/b^2=1的长轴为短轴的根号3倍,则 a=根号3b则椭圆方程变为x^2/3b^2+y^2/b^2=1C(根号3b,0)向量OC=(根号3b,0) 联立椭圆方程与直线方程,x^2/3b^2+y^2/b^2=1y=x得 x^2/3b^2+x^2/b^2=1解得 x=√3b/2所以 A(√3b/2,√3b/2)向量OA=(根号3b/2,根号3b/2)向量OA*向量OC=3b^2/2所以 3b^2/2 =3/2b^2=1 a^2=3椭圆的方程为 x^2/3+y^2=1