First slide

Equation of line in Straight lines

Question

Let a,b,c be in A.P. and $x, y, z$ be in G.P.. Then the points $(a, x), (b, y)$ and $(c, z)$ will be collinear if

Moderate

A

$x^{2} = y$
B

$x = y = z$
C

$y^{2} = z$
D

$x = z^{2}$

Solution

Given that $a, b, c$ are in A.P.
$\Rightarrow a - b = b - c$
Now, $(a, x), (b, y)$ and $(c, z)$ are collinear.
$\Rightarrow \frac{x - y}{a - b} = \frac{y - z}{b - c}$
$\Rightarrow \frac{x - y}{y - z} = \frac{a - b}{b - c} = 1$
$\Rightarrow x - y = y - z$
So, x,y are in A.P.
Thus x,y,z are in A.P. and also in G.P,
$∴ x = y = z$

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