If A and B are (-2, -2) and (2, -4) respectively, find the coordinates

Leave a Comment / coordinate geometry / By Sapling Academy

Q) If A and B are (-2, -2) and (2, -4) respectively, find the coordinates of P such that AP = $\frac{3}{7}$ AB and P lies on the line segment AB.

Ans:

Let the coordinates of P are (X, Y)

Since P lies on the line AP,

∴ AP + PB = AB … (1)

Given that AP = $\frac{3}{7}$ AB

∴ 7 AP = 3 AB …. (2)

Substituting value of AB from equation (1) , we get:

7 AP = 3 (AP + PB)

∴ 7 AP = 3 AP + 3 PB

∴ 4 AP = 3 PB

∴ $\frac{AP}{PB} = \frac{3}{4} = \frac{m_1}{m_2}$ … (3)

(m₁and m₂are the factors of division of the line)

Now, by section formula, coordinates of point P (X, Y) which lies between two points (x₁, y₁), (x₂, y₂) will be given by:

P (X,Y) = $(\frac{m_1 \times_2 + m_2 \times_1}{m_1 + m_2}, \frac{m_1 Y_2 + m_2 Y_1}{m_1 + m_2})$

Coordinates of the line segment AB are: (-2, -2) and (2, -4)

and from equation (3), we have values of m₁ and m₂

By substituting these values in above section formula, we get:

P (X,Y) = $(\frac{3 \times 2 + 4 \times (- 2)}{3 + 4}, \frac{3 \times (- 4) + 4 \times -2}{3 + 4})$

P (X,Y) = $(\frac{6 - 8}{7}, \frac{- 12 - 8}{7})$

P (X,Y) = $(\frac{- 2}{7}, \frac{- 20}{7})$

Therefore, the coordinates of point P are $(\frac{- 2}{7}, \frac{- 20}{7})$

Leave a Comment Cancel Reply