Question

The point P(1, 2) divides the join of A(–2, 1) and B(7, 4) are in the ratio of

(a) 3 : 2 (b) 2 : 3 (c) 2 : 1 (d) 1 : 2​

Answer 1

$Let \: P(1,2)\: divide\: AB \: internally \: in \: the \\ratio \: k : 1$

$Here, A(-2,1) = (x_{1} , y_{1}) \: and \\B(7,4) = (x_{2} , y_{2})$

/* Using Section Formula */

$P = \Big( \frac{kx_{2} + x_{1}}{k+1} , \frac{ky_{2} + y_{1}}{k+1}\Big)$

$\implies P(1,2) = \Big( \frac{k\times 7 + (-2)}{k+1} , \frac{ k \times 4 + 1 }{k+1}\Big)$

$= \Big( \frac{7k-2}{k+1} , \frac{4k+1}{k+1}\Big)$

$\implies 1 = \frac{7k-2}{k+1}$

$\implies k + 1 = 7k - 2$

$\implies 1 + 2 = 7k - k$

$\implies 3 = 6k$

$\implies \frac{3}{6} = k$

$\implies \frac{1}{2} = \frac{k}{1}$

Therefore.,

$\red{ Required \: ratio ( k : 1 ) } \green { = 1 : 2 }$

$Option \: \blue { ( d ) } \: is \: correct$

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