Statement-1 (A): If c is a variable, then the centroids of the tr… - Vidyadip

MCQ

Statement-1 (A): If $c$ is a variable, then the centroids of the triangles having vertices at $P(1, a), Q(c, b)$, and $R\left(c^2, 1\right)$ will never lie on $y$-axis.
Statement-2 (R) : If $x$-coordinate of a point is not zero, it does not lie on $y$-axis.

✓
Statement-1 and Statement-2 are True; Statement- 2 is a correct explanation for Statement-1.
B
Statement -1 and Statement- 2 are True; Statement- 2 is not a correct explanation for Statement-1,
C
Statement-1 is True, Statement-2 is False.
D
Statement-1 is False, Statement- 2 is True.

✓

Answer

Correct option: A.

Statement-1 and Statement-2 are True; Statement- 2 is a correct explanation for Statement-1.

(A)Statement-1 and Statement-2 are True; Statement- 2 is a correct explanation for Statement-1.
The coordinates of the centroid of $\triangle P Q R$ are $\left(\frac{1+c+c^2}{3}, \frac{a+b+1}{3}\right)$.
Now, $\quad c^2+c+1=c^2+2\left(\frac{1}{2}\right) c+\left(\frac{1}{2}\right)^2-\left(\frac{1}{2}\right)^2+1=\left(c+\frac{1}{2}\right)^2+\frac{3}{4} \geq \frac{3}{4} \quad\left[\because\left(c+\frac{1}{2}\right)^2 \geq 0\right.$ forall $\left.c\right]$
Thus, $\quad 1+c+c^2 \neq 0$ for any $c \Rightarrow \frac{1+c+c^2}{3} \neq 0$ for any $c$
Thus, $x$-coordinate of the centroid of $\triangle P Q R$ cannot be zero for any value of $c$. Consequently, centroid of $\triangle P Q R$ does not lie on $y$-axis. So, statment- 1 is true.
Statement-2 is also true and it is a correct explanation for statement-1. Hence, option (a) is correct.

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