Question

If the point  \[C \left( - 1, 2 \right)\] divides internally the line segment joining the points  A (2, 5)  and B( x, y ) in the ratio 3 : 4 , find the value of x² + y² .

 

Answer 1

It is given that the point C(–1, 2) divides the line segment joining the points A(2, 5) and B(x, y) in the ratio 3 : 4 internally.
Using the section formula, we get

\[\left( - 1, 2 \right) = \left( \frac{3 \times x + 4 \times 2}{3 + 4}, \frac{3 \times y + 4 \times 5}{3 + 4} \right)\]

\[ \Rightarrow \left( - 1, 2 \right) = \left( \frac{3x + 8}{7}, \frac{3y + 20}{7} \right)\]

\[ \Rightarrow \frac{3x + 8}{7} = - 1\text{ and }  \frac{3y + 20}{7} = 2\]

⇒ 3x + 8 = –7 and 3y + 20 = 14
⇒ 3x = –15 and 3y = –6
⇒ x = –5 and y = –2
∴ x² + y² = 25 + 4 = 29
Hence, the value of x² + y² is 29.