Question

A₁ (x₁, y₁), A₂ (x₂, y₂), A₃ (x₃, y₃) and A₄ (x₄, y₄) are 4 points, p₁ divides A₁A₂ in the ratio 1 ∶ 1, p₂ divides P₁A₃ in the ratio 1 ∶ 2, P₃ divides p₂A₄ in the ratio 1 ∶ 3. Then the coordinates of p₃ are:
1. \(\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}\right)\)
2. \(\left(\dfrac{x_1+x_4}{2}, \dfrac{y_2+y_4}{2}\right)\)
3. \(\left(\dfrac{x_1+x_2+x_3}{3}, \dfrac{y_1+y_2+y_3}{3}\right)\)
4. \(\left(\dfrac{x_1+x_3}{2}, \dfrac{y_1+y_3}{2}\right)\)

Answer 1

Correct Answer - Option 1 : \(\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}\right)\)

Concept:

Section formula is used to find the ratio in which a line segment is divided by a point internally or externally. According to this formula, if a point P (lying on AB) divides AB in the ratio m : n then,

\(P = \left ( \frac{mx_{2}+nx_{1}}{m+n},\, \frac{my_{2}+ny_{1}}{m+n} \right )\)      ----(1)

Calculation:

Since p1 divides A1A2 in the ratio 1 ∶ 1, then the coordinates of p₁ are given by-

p₁ = \(\left ( \frac{x_{2}+x_{1}}{2},\, \frac{y_{2}+y_{1}}{2} \right )\)      [using (1)]      ----(2)

Similarly, since p2 divides p1A3 in the ratio 1 ∶ 2, then the coordinates of p₂ are given by-

p₂ = \(\left ( \frac{1}{3}\left ( x_{3}+\frac{2(x_{1}+x_{2})}{2} \right ),\, \frac{1}{3}\left ( y_{3}+\frac{2(y_{1}+y_{2})}{2} \right ) \right )\)

⇒ p₂ = \(\left ( \frac{x_{1}+x_{2}+x_{3}}{3},\, \frac{y_{1}+y_{2}+y_{3}}{3} \right )\)

Again, since p3 divides p₂A₄ in the ratio 1 ∶ 3, then the coordinates of p₃ are given by-

p₃ = \(\left ( \frac{1}{4}\left ( x_{4}+\frac{3(x_{1}+x_{2}+x_{3})}{3} \right ),\, \frac{1}{4}\left ( y_{4}+\frac{3(y_{1}+y_{2}+y_{3})}{3} \right ) \right )\)

⇒ p₃ = \(\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}\right)\) 

Hence, the coordinates of p₃ are \(\left(\dfrac{x_1+x_2+x_3+x_4}{4}, \dfrac{y_1+y_2+y_3+y_4}{4}\right)\)