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
$$\begin{gathered} \left(-1,2\right)=\left(\frac{3\times x+4\times 2}{3+4},\frac{3\times y+4\times 5}{3+4}\right) \\ \Rightarrow \left(-1,2\right)=\left(\frac{3x+8}{7},\frac{3y+20}{7}\right) \\ \Rightarrow \frac{3x+8}{7}=-1\mathrm{and}\frac{3y+20}{7}=2 \end{gathered}$$
⇒ 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.