I : The ratio in which L=ax+by+c=0 divides the line segment joining A(x1,y1) and B(x2,y2) is −L1L2( where Li=axi+byi+c)
II: The equation of the line in which (x1,y1) divides the line segment between the coordinate axes in the ratio m:n is nxx1+myy1=m+n. Then which of the following is true?
P lie on L=ax+by+c=0, So,
a(kx1+x2)k+1+b(ky1+y2)k+1+c=0
k(ax1+by1+c)+ax2+by2+c=0
k=(−ax2+by2+c)(ax1+by1+c)=−L2L1
Ratio in which L=ax+by+c=0 divides the line segment joining A(x1,y1)B(x2,y2) is −L2L1 True
II. (x1,y1) divides the line segment joining (x,0) &(0,y) in the ratio m:n
So, x1=nxm+n & y1=mym+n
x=x1(m+n)n & y=y1(m+n)m
So, equation of AB is y=−y1xn1m(x−x1(m+n)n)
−myx1=ny1x−n1y1m−x1y1x
⇒hxx1+myy1=m+n True