Geometry

Step 1: The area of a triangle with vertices is given by the formula: Substituting the coordinates into this formula: Step 2: Simplifying the expression: Step 3: We are given that the area is 3 square units, so: Step 4: Solving the absolute value equation: For : For : Step 5: Therefore, the possible values of are and .

Step 1: Understanding the Concept:
The slope-intercept form of a line is , where is the slope and is the y-intercept.
Step 2: Detailed Explanation:
1. Given slope ( ) = 2. 2. The line passes through the origin , which means the y-intercept ( ) is 0. 3. Substituting these into the formula: . 4. Result: .
Step 3: Final Answer:
The correct option is (a).

Step 1: Understanding the Concept:
The distance between two points and is found using the distance formula: .
Step 2: Detailed Explanation:
Let and . $ $
Step 3: Final Answer:
The correct option is (c).

Concept: Area of triangle:
Step 1: Substitute points.
Step 2:
Step 3:

Concept: For modulus functions: Area is always taken positive.
Step 1: Split the interval.
Step 2: Evaluate integrals.
Step 3:

Concept: Standard equation of ellipse: Area of full ellipse =
Step 1: Convert to standard form. So,
Step 2: Find total area.
Step 3: First quadrant area.