Question

直线x－y＋1＝0被椭圆x²÷16＋y²÷4＝1截得的弦长为多少,

Answer 1

x－y＋1＝0

y=x+1代入x²÷16＋y²÷4＝1

x²÷16＋(x+1)²÷4＝1

x²＋4(x+1)²＝16

5x²+8x-12=0

根据韦达定理：x1+x2=-8/5,x1x2=-2/5

(x2-x1)² = (x2+x1)² -4x1x2 = (8/5)²+412/5 = 304/25

(y2-y1)² = (x2+1-x1-1)² = (x2-x1)² = 304/25

截得的弦长 = √{(x2-x1)²+(y2-y1)²} = √{608/5) = 4√38/5