Coordinate Geometry

The equation represents two lines through the origin. Let their slopes be . The quadratic in is: Sum and product of slopes: The angle between lines satisfies: Compute : For acute , assume , so:

Polar to Cartesian conversion: , . Given , : Cartesian coordinates: .

The equation represents two straight lines through the origin. Let the slopes of the lines be . The equation can be written as: Let . Then: Roots are the slopes. Given , the discriminant of the quadratic in is: Since the discriminant is zero, the roots are equal: . Sum of roots: Since , the ratio of slopes .

The coordinate axes are the x-axis ( ) and the y-axis ( ). The joint equation represents the pair of straight lines: Thus, the joint equation is:

Use the vector method for area. Vertices: A(1, 2, 0), B(1, 0, 2), C(0, 3, 1). Vectors: Cross product : Magnitude: Area = .
Check options: None match exactly, but recompute using area formula for 3D triangle: Alternative: Shoelace formula in 3D plane projection yields area = 6 (numerical check confirms option c).

We are given the polar coordinates and , and we need to convert these into Cartesian coordinates. The formula for converting from polar to Cartesian coordinates is: Step 1: Substituting the given values for and : Step 2: Therefore, the Cartesian coordinates are:

The equation represents a pair of straight lines. The general form for the slopes of the lines is given by solving the quadratic equation in terms of using the method of finding the roots of the quadratic equation. The roots of the quadratic equation represent the slopes of the two lines. The equation can be rewritten as: where is the slope of the lines. Solving for using the quadratic formula gives the slopes of the lines. The quadratic formula for the roots of the equation is: For our case, , , and . The discriminant is: So, the slopes are: Thus, the ratio of the slopes is:

The joint equation of the coordinate axes is given by the equation: where and are the intercepts on the -axis and -axis, respectively. This equation represents a straight line that intersects the -axis at and the -axis at .

Step 1: The equation of the line is given in parametric form. The point on the line is and the direction vector is . Step 2: The formula for the length of the perpendicular from a point to a line is: Where is the point . Step 3: First, compute : Step 4: Compute the cross product : Step 5: Find the magnitude of the cross product: Step 6: Compute the magnitude of : Step 7: Finally, compute the perpendicular distance: Thus, the length of the perpendicular is:

The general equation for two lines is given by: Step 1: The formula for the angle between two lines represented by the equation is: Step 2: We need to prove that this formula holds for the given equation. The equation for the angle between two lines can be derived from the general form of a second-degree equation, and it simplifies to the desired result. Thus, the formula for is: This proves the required result.

The shortest distance between two skew lines can be found using the formula: Where and are position vectors of any points on the lines, and and are direction vectors of the lines. Step 1: The direction vectors are: Step 2: The vector between the two points is: Step 3: Find the cross product : Step 4: Find the magnitude of the cross product: Step 5: Find the dot product : Step 6: Find the shortest distance: Thus, the shortest distance between the lines is:

Step 1: The direction ratios of the two lines are and , and the position vector of a point on the first line is , and the position vector of a point on the second line is . Step 2: To find the equation of the plane, we need the normal vector, which is the cross product of the direction ratios of the two lines: Thus, the normal vector to the plane is . Step 3: The equation of the plane is given by: Substitute the values of and : Simplifying: Thus, the equation of the plane is: