直线y=kx+m(m,k>0)与椭圆x²/4+y²=1交于A,B两点,当|AB|=2,S△AOB=1时,求直线AB的方程.
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![直线y=kx+m(m,k>0)与椭圆x²/4+y²=1交于A,B两点,当|AB|=2,S△AOB=1时,求直线AB的方程.](/uploads/image/z/5301861-69-1.jpg?t=%E7%9B%B4%E7%BA%BFy%3Dkx%2Bm%28m%2Ck%3E0%29%E4%B8%8E%E6%A4%AD%E5%9C%86x%26%23178%3B%2F4%2By%26%23178%3B%3D1%E4%BA%A4%E4%BA%8EA%2CB%E4%B8%A4%E7%82%B9%2C%E5%BD%93%7CAB%7C%3D2%2CS%E2%96%B3AOB%3D1%E6%97%B6%2C%E6%B1%82%E7%9B%B4%E7%BA%BFAB%E7%9A%84%E6%96%B9%E7%A8%8B.)
直线y=kx+m(m,k>0)与椭圆x²/4+y²=1交于A,B两点,当|AB|=2,S△AOB=1时,求直线AB的方程.
直线y=kx+m(m,k>0)与椭圆x²/4+y²=1交于A,B两点,当|AB|=2,S△AOB=1时,求直线AB的方程.
直线y=kx+m(m,k>0)与椭圆x²/4+y²=1交于A,B两点,当|AB|=2,S△AOB=1时,求直线AB的方程.
|AB|=2,S△AOB=1,则原点与直线kx - y + m = 0的距离为h = m/√(k² + 1)
S△AOB = (1/2)|AB|*h= (1/2)*2*h = h = 1
m/√(k² + 1) = 1,m² = k² + 1 (1)
y = kx + m代入椭圆的方程并整理:(4k² + 1)x² + 8kmx + 4(m² - 1) = 0
x₁ + x₂ = -8km/(4k² + 1)
x₁x₂ = 4(m² - 1)/(4k² + 1)
|AB|² = (x₁ - x₂)² + (y₁ - y₂)² = (x₁ - x₂)² + (kx₁ +m - kx₂ - m)²
= (k² + 1)(x₁ - x₂)²
= (k² + 1)[(x₁ + x₂)² - 4x₁x₂]
= (k² + 1)[(-8km)²/(4k² + 1)² - 4*4(m² - 1)/(4k² + 1)]
= 16(k² + 1)(4k² - m² + 1)/(4k² + 1)²
= 16(k² + 1)(4k² - k² - 1 + 1)/(4k² + 1)²
= 48k²(k² + 1)/(4k² + 1)² = 4
(2k² - 1)² = 0
k = √2/2
m = √6/2