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Question

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?

A
only I
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B
only II
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C
both I&II
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D
neither I nor II
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Solution

The correct option is C both I&II

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(xx1(m+n)n)

myx1=ny1xn1y1mx1y1x

hxx1+myy1=m+n True


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