First slide

Equation of line in Straight lines

Question

If $x_{1}, x_{2}, x_{3}$ as well as $y_{1}, y_{2}, y_{3}$ are in GP with same common ratio, then the points $P (x_{1}, y_{1}), Q (x_{2}, y_{2})$ and $R (x_{3}, y_{3})$

Moderate

A

lie on a straight line
B

lie on an ellipse
C

lie on a circle
D

are vertices of a triangle

Solution

Let $r$ be the common ratio. Then,
$x_{2} = x_{1} r, x_{3} = x_{1} r^{2}, y_{2} = y_{1} r$ and $y_{3} = y_{1} r^{2}$
$∴ \frac{y_{2} - y_{1}}{x_{2} - x_{1}} = \frac{y_{1} r - y_{1}}{x_{1} r - x_{1}} = \frac{y_{1}}{x_{1}}$
and, $\frac{y_{3} - y_{2}}{x_{3} - x_{2}} = \frac{y_{1} r^{2} - y_{1} r}{x_{1} r^{2} - x_{1} r} = \frac{y_{1}}{x_{1}}$
So,
Slope of $P Q =$ Slope of $Q R$
Hence, points $P, Q, R$ are collinear.

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