The planes: 2x - y + 4z = 5 and 5x - 2.5y + 1 | vedclass

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(D) The equations of the planes are:$2x - y + 4z = 5$  $(1)$$5x - 2.5y + 10z = 6$  $(2)$For two planes $a_1x + b_1y + c_1z = d_1$ and $a_2x + b_2y + c

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Parallel

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(D) The equations of the planes are:$2x - y + 4z = 5$  $(1)$$5x - 2.5y + 10z = 6$  $(2)$For two planes $a_1x + b_1y + c_1z = d_1$ and $a_2x + b_2y + c_2z = d_2$,they are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$.Calculating the ratios of the coefficients of $x, y,$ and $z$:$\frac{a_1}{a_2} = \frac{2}{5} = 0.4$$\frac{b_1}{b_2} = \frac{-1}{-2.5} = \frac{1}{2.5} = 0.4$$\frac{c_1}{c_2} = \frac{4}{10} = 0.4$Since $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} = 0.4$,the normal vectors of the planes are proportional.Also,check if the planes are coincident: $\frac{d_1}{d_2} = \frac{5}{6} 
eq 0.4$. Since the ratios of the coefficients are equal but not equal to the ratio of the constants,the planes are parallel and distinct.Therefore,the correct answer is $D$.

The planes: $2x - y + 4z = 5$ and $5x - 2.5y + 10z = 6$ are

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