Question

if the mid point of the line segment joining the points A(4a, 6b) and B(-10a, 4b) is M(3, 5), find the co-ordinates of A and B.

Answer 1

Given: Mid - point of the line segment joined by the points A(4a, 6b) and B(-10a, 4b) is M(3, 5).

To find: The coordinates of A and B.

Answer:

Mid - point formula:

$\tt \bigg(\dfrac{x_1\ +\ x_2}{2},\ \dfrac{y_1\ +\ y_2}{2}\bigg)$

From the given points, we have:

$\tt x_1\ =\ 4a\\\\x_2\ =\ -10a\\\\y_1\ =\ 6b\\\\y_2\ =\ 4b$

Using them in the formula,

$\tt \bigg(3,\ 5\bigg)\ =\ \bigg(\dfrac{4a\ +\ -10a}{2},\ \dfrac{6b\ +\ 4b}{2}\bigg)\\\\\\\bigg(3,\ 5\bigg)\ =\ \bigg(\dfrac{-6a}{2},\ \dfrac{10b}{2}\bigg)\\\\\\\bigg(3,\ 5\bigg)\ =\ \bigg(-3a,\ 5b\bigg)$

Equating the x coordinates,

3 = - 3a

-1 = a

Equating the y coordinates,

5 = 5b

1 = b

Therefore, M(3, 5) is the mid - point of the line segment joining the points A(-4, 6) and B(10, 4).

Answer 2

AnswEr :

Given,

• The point M(3,5) is the mid point of AB

• Coordinates of A and B are (4a,6b) and (-10a,4b) respectively

To finD : Values of a and b

Using the mid point formula,

$\sf \: \bigg(3 ,\: 5 \bigg) = \bigg( \dfrac{ - 10a + 4a}{2} , \dfrac{6b + 4b}{2} \bigg)$

WKT,

If A(x,y) = B(p,q) then x = p and y = q

Now,

$\sf \: - 10a + 4a = 6 \\ \\ \longrightarrow \: \sf \: - 6a = 6 \\ \\ \longrightarrow \: \boxed{ \boxed{ \sf \: a = - 1}}$

Also,

$\sf \: 6b + 4b = 10 \\ \\ \longrightarrow \: \boxed{ \boxed{ \sf \: b = 1}}$

Therefore,the values of a and b are - 1 and 1 respectively

Now,

• Coordinates of A = (-4,6)

• Coordinates of B = (10,4)