Progression (Sequence and Series)HardBloom L3
Question
If $a,b,c$ form an A.P. with comon difference $d( \neq 0)$ and $x,y,z$ form a G.P. with common ratio $r( \neq 1)$, then the area of triangle with vertices $(a,x),(b,y)$ and $(c,z)$ is independent of
Options
A.a
B.d
C.$x$
D.r
Solution
Sol. [Key Concept: Area of a triangle with A.P./G.P. coordinates via the determinant; check which parameter drops out.]
Area $= \frac{1}{2}\left| \begin{matrix} a & x & 1 \\ a + d & xr & 1 \\ a + 2d & xr^{2} & 1 \end{matrix} \right| = \frac{1}{2}xd(r - 1)^{2}$
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