95
Prove that the point P dividing the line segment joining the points A($-1, 7$) and B($4, -3$) in the ratio $3:2$, lies on the line $x - 3y = -1$. Also find length of PA and PB.
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Sol.
AP: PB = $3:2$
Coordinates of P = $(\frac{3\times 4+2\times (-1)}{3+2}, \frac{3\times (-3)+2\times 7}{3+2}) = (2,1)$ (1 Mark)
Substituting $x = 2$ and $y = 1$ in the given equation
L. H. S. = $x - 3y$
= $2 - 3(1)$
= $-1$ = R. H. S.
$\therefore$ P lies on the given line (1 Mark)
PA = $\sqrt{(2 + 1)^2 + (1 - 7)^2} = \sqrt{45}$ or $3\sqrt{5}$ (1/2 Mark)
PB = $\sqrt{(2 - 4)^2 + (1 + 3)^2} = \sqrt{20}$ or $2\sqrt{5}$ (1/2 Mark)
AP: PB = $3:2$
Coordinates of P = $(\frac{3\times 4+2\times (-1)}{3+2}, \frac{3\times (-3)+2\times 7}{3+2}) = (2,1)$ (1 Mark)
Substituting $x = 2$ and $y = 1$ in the given equation
L. H. S. = $x - 3y$
= $2 - 3(1)$
= $-1$ = R. H. S.
$\therefore$ P lies on the given line (1 Mark)
PA = $\sqrt{(2 + 1)^2 + (1 - 7)^2} = \sqrt{45}$ or $3\sqrt{5}$ (1/2 Mark)
PB = $\sqrt{(2 - 4)^2 + (1 + 3)^2} = \sqrt{20}$ or $2\sqrt{5}$ (1/2 Mark)