Question

Find the angle between the straight lines joining the origin to the point of intersection of

3x²+5xy−3y²+2x+3y=0 and 3x−2y=1

Answer 1

To find the angle between straight lines from the origin to an intersection point, solve the equations for the point's coordinates, form vectors from the origin to this point, and use the vector angle formula to calculate the angle.

Step-by-step explanation:

To find the angle between the straight lines originating from the origin and intersecting at a point defined by the intersection of 3x²+5xy−3y²+2x+3y=0 and 3x−2y=1, we need to follow these steps:

1. Solve the two given equations to find the coordinates of the point of intersection.
2. With these coordinates, form two vectors representing the lines from the origin to the intersection point.
3. Use the formula for the angle between two vectors Angle = cos⁻¹((Ax Bx + Ay By + Az Bz) ⁄ (A*B)), where A and B are the magnitudes of the vectors and Ax, Ay, Bx, By are the vector components along the x- and y-axes.
4. Calculate the magnitude of each vector using A = √(4x²+A₁²) and the angle using 0 = tan⁻¹(Ay⁄A₁).
5. Find the components of each vector, Ax = A cos 0 and Ay = A sin 0, along the chosen perpendicular axes.

Using these steps, we can determine the angle between the lines.