The midpoint P of the line segment joining the points A (-10, 4) and B (-2, 0) lies on the line segment joining the points C (-9, -4) and D (-4, y). Find the ratio in which P divides CD and also, find the value of y.
Answer
361.8k+ views
Hint: Find the coordinates of the midpoint P of the line segment AB. Then use the section formula of line segment CD for the abscissa of point P to find the ratio in which P divides CD. Use the section formula of line segment CD for the ordinate of point P to find y.
Complete step-by-step answer:
From section formula, if P (x, y) divides the line segment joining \[C({x_3},{y_3})\] and \[D({x_4},{y_4})\] in the ratio m:n, then:
\[x = \dfrac{{m{x_4} + n{x_3}}}{{m + n}};{\text{ }}y = \dfrac{{m{y_4} + n{y_3}}}{{m + n}}{\text{ }}.........{\text{(2)}}\]
Substituting equation (1) in equation (2) and using coordinates of C and D, we get:
\[ - 6 = \dfrac{{m( - 4) + n( - 9)}}{{m + n}}{\text{ }}..........{\text{(3)}}\]
\[ 2 = \dfrac{{m(y) + n( - 4)}}{{m + n}}{\text{ }}...........{\text{(4)}}\]
Simplifying equation (3) to get the ratio in which P divided CD, we get:
\[ - 6 = \dfrac{{ - 4m - 9n}}{{m + n}}{\text{ }}\]
\[ - 6(m + n) = - 4m - 9n\]
\[ - 6m - 6n = - 4m - 9n\]
\[ - 6m + 4m = - 9n + 6n\]
\[ - 2m = - 3n\]
\[\dfrac{m}{n}{\text{ = }}\dfrac{3}{2}{\text{ }}..........{\text{(5)}}\]
Simplifying equation (4) to obtain the value of y, we get:
\[2 = \dfrac{{my - 4n}}{{m + n}}\]
\[2(m + n) = my - 4n\]
\[2m + 2n = my - 4n\]
Gathering terms containing m on RHS and terms containing n on LHS, we get:
\[4n + 2n = my - 2m\]
\[6n = m(y - 2)\]
Divide both sides by n, to get:
\[6 = \dfrac{m}{n}(y - 2)\]
Substituting equation (5) in the above equation, we get:
\[6 = \dfrac{3}{2}(y - 2)\]
Multiply both sides by \[\dfrac{2}{3}\] and simplify.
\[\dfrac{2}{3} \times 6 = y - 2\]
\[4 = y - 2\]
\[y = 6\]
Hence, the value of y is 6
Therefore, P divides CD in the ratio 3:2 and the value of y is 6.
Note: The possibility for mistake is writing the section formula for points \[C({x_3},{y_3})\] and \[D({x_4},{y_4})\] wrongly as \[x = \dfrac{{m{x_3} + n{x_4}}}{{m + n}};{\text{ }}y = \dfrac{{m{y_3} + n{y_4}}}{{m + n}}\] instead of \[x = \dfrac{{m{x_4} + n{x_3}}}{{m + n}};{\text{ }}y = \dfrac{{m{y_4} + n{y_3}}}{{m + n}}\] . You might also think, it is impossible to find three variables from two equations but you are just finding the ratio between m and n and then the value of y, which requires only two equations.
Complete step-by-step answer:

From section formula, if P (x, y) divides the line segment joining \[C({x_3},{y_3})\] and \[D({x_4},{y_4})\] in the ratio m:n, then:
\[x = \dfrac{{m{x_4} + n{x_3}}}{{m + n}};{\text{ }}y = \dfrac{{m{y_4} + n{y_3}}}{{m + n}}{\text{ }}.........{\text{(2)}}\]
Substituting equation (1) in equation (2) and using coordinates of C and D, we get:
\[ - 6 = \dfrac{{m( - 4) + n( - 9)}}{{m + n}}{\text{ }}..........{\text{(3)}}\]
\[ 2 = \dfrac{{m(y) + n( - 4)}}{{m + n}}{\text{ }}...........{\text{(4)}}\]
Simplifying equation (3) to get the ratio in which P divided CD, we get:
\[ - 6 = \dfrac{{ - 4m - 9n}}{{m + n}}{\text{ }}\]
\[ - 6(m + n) = - 4m - 9n\]
\[ - 6m - 6n = - 4m - 9n\]
\[ - 6m + 4m = - 9n + 6n\]
\[ - 2m = - 3n\]
\[\dfrac{m}{n}{\text{ = }}\dfrac{3}{2}{\text{ }}..........{\text{(5)}}\]
Simplifying equation (4) to obtain the value of y, we get:
\[2 = \dfrac{{my - 4n}}{{m + n}}\]
\[2(m + n) = my - 4n\]
\[2m + 2n = my - 4n\]
Gathering terms containing m on RHS and terms containing n on LHS, we get:
\[4n + 2n = my - 2m\]
\[6n = m(y - 2)\]
Divide both sides by n, to get:
\[6 = \dfrac{m}{n}(y - 2)\]
Substituting equation (5) in the above equation, we get:
\[6 = \dfrac{3}{2}(y - 2)\]
Multiply both sides by \[\dfrac{2}{3}\] and simplify.
\[\dfrac{2}{3} \times 6 = y - 2\]
\[4 = y - 2\]
\[y = 6\]
Hence, the value of y is 6
Therefore, P divides CD in the ratio 3:2 and the value of y is 6.
Note: The possibility for mistake is writing the section formula for points \[C({x_3},{y_3})\] and \[D({x_4},{y_4})\] wrongly as \[x = \dfrac{{m{x_3} + n{x_4}}}{{m + n}};{\text{ }}y = \dfrac{{m{y_3} + n{y_4}}}{{m + n}}\] instead of \[x = \dfrac{{m{x_4} + n{x_3}}}{{m + n}};{\text{ }}y = \dfrac{{m{y_4} + n{y_3}}}{{m + n}}\] . You might also think, it is impossible to find three variables from two equations but you are just finding the ratio between m and n and then the value of y, which requires only two equations.
Last updated date: 27th Sep 2023
•
Total views: 361.8k
•
Views today: 3.61k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE

Drive an expression for the electric field due to an class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

What is the past tense of read class 10 english CBSE
