Anonymous asked in 科學數學 · 2 decades ago

二個數學證明 要用英文喔 二十點

suppose that (a,b)is on the circle x平方加Y平方等於r平方 show that the line

ax+by=r平方 is tangent to the circle at (a,b)

question2:show that the equation of the line with x-intercept a不等於0 and y-intercept

b不等於0 can be written as x/a+y/b=1

1 Answer

  • dd
    Lv 6
    2 decades ago
    Favorite Answer

    Q1let L be the line tangent to the circle C: x2+y2=r2and let P be the point (a,b)and let O be the origin (0,0)and let m1 and m2 be the slopes of OP and L respectivelythen, we have that the line OP is perpendicular to Lso, m1*m2=-1 i.e. (b-0)/(a-0)*m2=-1m2=-a/blet (x,y) be a point on L(y-b)/(x-a)=m2=-a/ba(x-a)+b(y-b)=0ax+by=a2+b2but P is on Cso, a2+b2=r2then L is ax+by=a2+b2=r2so, we provedQ2let (x,y) be a point on line Lbecause the line L with x-intercept a≠0 and y-intercept b≠0, L pass through (a,0) and (0,b)so, we have (y-0)/(x-a) = (y-b)/(x-0)=> xy=(x-a)(y-b)=> xy=xy-bx-ay+ab=> bx+ay=ab=> x/a+y/b=1so, we proved

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