A line segment joining A(-1, $\frac{5}{3}$ ) and B(a, 5) is divided in the ratio 1 : 3 at P, the point where the line segment AB intersects the y-axis.
(i) Calculate the value of 'a'.
(ii) Calculate the co-ordinates of 'P'.

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Solution

Since, the line segment AB intersects the y-axis at point P, let the co-ordinates of point P be (0, y).
P divides AB in the ratio 1: 3.

$left parenthesis 0 comma y right parenthesis equals open parentheses fraction numerator 1 cross times a plus 3 cross times left parenthesis negative 1 right parenthesis over denominator 1 plus 3 end fraction comma fraction numerator 1 cross times 5 plus 3 cross times begin display style 5 over 3 end style over denominator 1 plus 3 end fraction close parentheses left parenthesis 0 comma y right parenthesis equals open parentheses fraction numerator a minus 3 over denominator 4 end fraction comma 10 over 4 close parentheses 0 equals fraction numerator a minus 3 over denominator 4 end fraction space space space space space a n d space space space y equals 10 over 4 a equals 3 space space space space space space space space space space space space space a n d space space space space y equals 5 over 2 equals 2 1 half t h u s space comma t h e space v a l u e space o f space a space i s space 3 space a n d space t h e space c o minus o r d i n a t e s space o f space P space a r e space open parentheses 0 comma 2 1 half close parentheses$

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