Question

GET BRAINLIEST FOR THE CORRECT ANSWER AND EXPLANATION!

Answer 1

Interpreting the inequality, it is found that the correct option is given by F.

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• The first equation is of the line.
• The equal sign is present in the inequality, which means that the line is not dashed, which removes option G.

In standard form, the equation of the line is:

$x + 2y = 6$

$2y = 6 - x$

$y = -0.5x + 2$

Thus it is a decreasing line, which removes options J.

• We are interested in the region on the plane below the line, that is, less than the line, which removes option H.

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• As for the second equation, the normalized equation is:

$3x^2 + 3y^2 = 12$

$3(x^2 + y^2) = 12$

$x^2 + y^2 = 4$

• Thus, a circle centered at the origin and with radius 2.
• Now, we have to check if the line $x + 2y - 6 = 0$, with coefficients $a = 1, b = 2, c = -6$, intersects the circle, of centre $x = 0, y = 0$
• First, we find the following distance:

$d = \sqrt{\frac{|ax + by + c|}{a^2 + b^2}}$

• Considering the coefficients of the line and the center of the circle.

$d = \sqrt{\frac{|1(0) + 2(0) - 6|}{1^2 + 2^2}} = \sqrt{\frac{6}{5}} = 1.1$

• This distance is less than the radius, thus, the line intersects the circle, which removes option K, and states that the correct option is given by F.

A similar problem is given at https://brainly.com/question/16505684