Question

Example 9: Find the ratio in which the y-axis divides the line segment joinings
points (5.-6) and (-1,-4). Also find the point of intersection.
fam
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Answer 1

☯ Let a point P (x,y) divides the line segment joining the point A (-5,2) and B (9,-2).

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》Let's consider the required ratio be m : n.

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$\underline{\bigstar\:\boldsymbol{Using\:Section\:Formula\::}}$

$\star\;{\boxed{\sf{\pink{(x,y) = \bigg( \dfrac{m x_2 + n x_1}{m + n}\:,\: \dfrac{m y_2 + n y_1}{m + n} \bigg)}}}}$

$\sf Here \begin{cases} & \sf{x_1\:,\: x_2 = \bf{5\:,\:-1}} \\ & \sf{y_1\:,\:y_2 = \bf{ - 6\:,\: - 4}} \end{cases}$

Given that,

• The given points are divided by y - axis.

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$\therefore$ It x - coordinate will be 0.

Therefore, Coordinates of P is (0,y).

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$\underline{\bigstar\:\boldsymbol{According\:to\:the\:question\::}}$

$:\implies\sf \dfrac{m x_2 + n x_1}{m + n} = 0\\ \\ \\ :\implies\sf \dfrac{m(-1) + n(5)}{m + n} = 0\\ \\ \\:\implies\sf -1 m + 5n = 0\\\\\\ :\implies\sf -1m = -5n\\\\\\ :\implies\sf \dfrac{m}{n} = \dfrac{5}{1}\\\\\\ :\implies{\underline{\boxed{\frak{\purple{m:n= 5:1}}}}}\;\bigstar$

$\therefore{\underline{\sf{The\:ratio\:in\:which\:y-axis\:divides\:the\:line\:segment\: {\textsf{\textbf{5:1}}}.}}}$

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》Now, finding y - coordinate of the point of division (P).

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$:\implies\sf \bigg( \dfrac{m x_2 + n x_1}{m + n}\:,\: \dfrac{m y_2 + n y_1}{m + n} \bigg)\\\\\\ :\implies\sf \bigg(0\;,\; \dfrac{m y_2 + n y_1}{m + n} \bigg)\\ \\ \\:\implies\sf \bigg(0\;,\; \dfrac{5 \times (-4) + 1 \times (-6)}{5+1} \bigg)\\ \\ \\:\implies\sf \bigg(0\;,\; \dfrac{-20 - 6}{6} \bigg)\\ \\ \\:\implies\sf \bigg(0\;,\; \dfrac{-26}{6} \bigg)\\ \\ \\:\implies{\underline{\boxed{\frak{\pink{\bigg(0\;,\; \dfrac{-13}{3} \bigg)}}}}}\;\bigstar$

$\therefore{\underline{\sf{Hence,\:the\:coordinates\:of\:point\:of\:intersection\:are\: \bf{\bigg(0\;,\; \dfrac{-13}{3} \bigg)}.}}}$

Answer 2

Answer:

Required Answer :-

Let us assume P (x,y) divides the line segment joining the point A (-5,2) and B (9,-2).

mx2 + nx1/ m + n= 0

m(-1) + n(5)/ m + n = 0

-1m + 5n = 0

-1m =0- 5n

-1m = -5n

m:n = 5:1

The ratio is 5:1

Now,

We are finding y - coordinate of the point of division (P).

(0,)(5(-4)+ 1(-6)/ 5+1)

0, -20 - 6/6

0,-26/6

Coordinates :- 0,-13/3