(1)焦點(diǎn)在X軸,一個(gè)頂點(diǎn)到兩個(gè)焦點(diǎn)的距離分別是7和1,則該頂點(diǎn)應(yīng)在X軸,
∴2c=7-1=6,c=6/2=3,長(zhǎng)半軸a=c+1=4,短半軸b=√(a^2-c^2)=√7,
橢圓方程為:x^2/16+y^2/7=1.
(2).|OP|/|OM|=λ,設(shè)M(x,y),P(x,k),P點(diǎn)與M橫坐標(biāo)相等,k是縱坐標(biāo),|OP|=√(x^2+k^2),
|OM|=√(x^2+y^2),P在橢圓上,x^2/16+k^2/7=1,
k=√112-7x^2)/4,x^2+(112-7x^2)/16=λ^2(x^2+y^2),點(diǎn)M的軌跡方程為:
x^2/(7(16λ^2-9)/16λ^2+y^2/7=1
當(dāng)λ>3/4時(shí),為橢圓,λ