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Question Number 24181 by gopikrishnan005@gmail.com last updated on 14/Nov/17

if the points C(−1,2) divides internally the line segment joining the points A(2,5) and B(x,y) in the ratio 3:4 find the value of x^2 +y^2

$${if}\:{the}\:{points}\:{C}\left(−\mathrm{1},\mathrm{2}\right)\:{divides}\:{internally}\:{the}\:{line}\:{segment}\:{joining}\:{the}\:{points}\:{A}\left(\mathrm{2},\mathrm{5}\right)\:{and}\:{B}\left({x},{y}\right)\:{in}\:{the}\:{ratio}\:\mathrm{3}:\mathrm{4}\:{find}\:{the}\:{value}\:{of}\:{x}^{\mathrm{2}} +{y}^{\mathrm{2}} \\ $$

Answered by mrW1 last updated on 14/Nov/17

((−1−2)/(x−2))=(3/(3+4))=(3/7) ⇒3(x−2)=7(−1−2) ⇒x=−5 ((2−5)/(y−5))=(3/(3+4))=(3/7) ⇒3(y−5)=7(2−5) ⇒y=−2 ⇒x^2 +y^2 =25+4=29

$$\frac{−\mathrm{1}−\mathrm{2}}{{x}−\mathrm{2}}=\frac{\mathrm{3}}{\mathrm{3}+\mathrm{4}}=\frac{\mathrm{3}}{\mathrm{7}} \\ $$$$\Rightarrow\mathrm{3}\left({x}−\mathrm{2}\right)=\mathrm{7}\left(−\mathrm{1}−\mathrm{2}\right) \\ $$$$\Rightarrow{x}=−\mathrm{5} \\ $$$$ \\ $$$$\frac{\mathrm{2}−\mathrm{5}}{{y}−\mathrm{5}}=\frac{\mathrm{3}}{\mathrm{3}+\mathrm{4}}=\frac{\mathrm{3}}{\mathrm{7}} \\ $$$$\Rightarrow\mathrm{3}\left({y}−\mathrm{5}\right)=\mathrm{7}\left(\mathrm{2}−\mathrm{5}\right) \\ $$$$\Rightarrow{y}=−\mathrm{2} \\ $$$$ \\ $$$$\Rightarrow{x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{25}+\mathrm{4}=\mathrm{29} \\ $$

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