Show that the points P (a, b + c), Q (b, c + a) and R (c, a + b) … - Vidyadip

Question

Show that the points $P (a, b + c), Q (b, c + a)$ and $R (c, a + b)$ are collinear.

✓

Answer

Let $\therefore P (a, b + c) = (x_1, y_1)$
$\therefore Q (b, c + a) = (x_2, y_2)$
$\therefore R (c, a + b) = (x_3, y_3)$
The points P, Q, R will be collinear if slope of PQ and QR is the same.
Slope of PQ =$\frac{y_2-y_1}{x_2-x_1}$
$=\frac{c+a-(b+c)}{b-a}$
$=\frac{c+a-b-c}{b-a}$
$=\frac{a-b}{b-a}$
$=\frac{-(b-a)}{b-a}$
$=-1$
Slope of QR $=\frac{y_3-y_2}{x_3-x_2}$
$=\frac{(a+b)-(c+a)}{c-a}$
$=\frac{a+b-c-a}{c-b}$
$=\frac{b-c}{c-b}$
$=\frac{-(c-b)}{c-b}$
$=-1$
Hence, the points $P, Q,$ and $R$ are collinear.

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