Angle Between Two Lines

Concept: 3D Geometry - Direction Cosines and Angle Between Lines.

Step 1: Express in terms of and . From the first given linear equation, , we can isolate . This gives us .

Step 2: Substitute into the second quadratic equation. Take the derived expression for and substitute it into the second equation: . This becomes .

Step 3: Expand and simplify to find the relationship between and . Expand the squared term: . Simplifying this yields , which is a perfect square. Thus, , leading directly to .

Step 4: Determine the proportional direction ratios. Since we established that , substitute this back into our equation for : . Therefore, the relationship between the direction cosines is . This means the direction ratios for the lines are proportional to .

Step 5: Calculate the angle between the lines. Because solving the system of equations yielded only one unique set of direction ratios , it means both lines possess the exact same direction cosines. When two lines have identical direction cosines, they represent parallel or coincident lines in 3D space.

Mathematically, the angle between two lines with direction ratios and is given by . Substituting our derived identical ratios gives .

Since , the angle must be either or . Looking at the provided options, is the correct match. $ $

Given the equation and is , we can determine the value of as follows: By comparing the equation with the general form of a conic section: we can identify the coefficients: For an ellipse, we know the condition is: Substitute the values of , , and into this condition: Since , the maximum possible value for is . Thus, the value of is less than .