Question

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

• A. $x^2=y$
• B. $x=2^2$
• C. $y^2=2$
• D. $x=y=z$

Answer 1

The correct option is D $x=y=z$

solution: $(a,x),(b,y),(c,z)$ are collinear so all three points will lie on same line

so, slope of line formed buy = slope of line formed by $(a,x),(b,y{)}^{}=(b,y),(c,z)$

$\Rightarrow \frac{y-x}{b-a}=\frac{z-y}{c-b}=\text{(1)}$

$a,b,c$ are in $A\cdot P$

so $2b=a+c$

$b-a=d$

$c-b=d$

$x,y,z$ are in $G\cdot P$ with common ratio ${}^{\prime }{r}^{\prime }$ so $y^2=xz$

Naw $,\frac{y-x}{b-a}=\frac{z-y}{c-b}$

$\Rightarrow y-x=z-y$

$\Rightarrow 2y=x+z$

$\Rightarrow 2rx=x+r^{2}x$

$\Rightarrow 2r=1+r^2$

$\Rightarrow r^2-2r+1=0$

$\Rightarrow r=1$

so $z=x{r}^2=x$

$y=xr=x$

i.e $x=y=z$

Ansuver: option (D)