Answer
Verified
423.3k+ 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.
Recently Updated Pages
Assertion The resistivity of a semiconductor increases class 13 physics CBSE
The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths
How do you arrange NH4 + BF3 H2O C2H2 in increasing class 11 chemistry CBSE
Is H mCT and q mCT the same thing If so which is more class 11 chemistry CBSE
What are the possible quantum number for the last outermost class 11 chemistry CBSE
Is C2 paramagnetic or diamagnetic class 11 chemistry CBSE
Trending doubts
State the differences between manure and fertilize class 8 biology CBSE
Why are xylem and phloem called complex tissues aBoth class 11 biology CBSE
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
Difference Between Plant Cell and Animal Cell
What would happen if plasma membrane ruptures or breaks class 11 biology CBSE
Give 10 examples for herbs , shrubs , climbers , creepers
What precautions do you take while observing the nucleus class 11 biology CBSE
What would happen to the life of a cell if there was class 11 biology CBSE
Change the following sentences into negative and interrogative class 10 english CBSE