Question
If the point $C(-1,2)$ divides internally the line segment joining the points $A(2,5)$ and $B(x, y)$ in the ratio $3: 4$, find the value of $x^2+y^2$.
✓
Answer
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
$(-1, 2)=\Big(\frac{3\times\text{x}+4\times2}{3+4},\frac{3\times\text{y}+4\times5}{3+4}\Big)$
$\Rightarrow\ (-1, 2)=\Big(\frac{3\text{x}+8}{7},\frac{3\text{y}+20}{7}\Big)$
$\Rightarrow\ \frac{3\text{x}+8}{7}=-1$ and $\frac{3\text{y}+20}{7}=2$
$\Rightarrow 3x + 8 = -7$ and $3y + 20 = 14$
$\Rightarrow 3x = -15$ and $3y = -6$
$\Rightarrow x = -5$ and $y = -2$
$\therefore x^2 + y^2 = 25 + 4 = 29$
Hence, the value of $x^2 + y^2$ is $29.$
Using the section formula, we get
$(-1, 2)=\Big(\frac{3\times\text{x}+4\times2}{3+4},\frac{3\times\text{y}+4\times5}{3+4}\Big)$
$\Rightarrow\ (-1, 2)=\Big(\frac{3\text{x}+8}{7},\frac{3\text{y}+20}{7}\Big)$
$\Rightarrow\ \frac{3\text{x}+8}{7}=-1$ and $\frac{3\text{y}+20}{7}=2$
$\Rightarrow 3x + 8 = -7$ and $3y + 20 = 14$
$\Rightarrow 3x = -15$ and $3y = -6$
$\Rightarrow x = -5$ and $y = -2$
$\therefore x^2 + y^2 = 25 + 4 = 29$
Hence, the value of $x^2 + y^2$ is $29.$
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