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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 .
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Solution
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.
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