List of top Mathematics Questions on Family of Lines asked in AP EAMCET

If the pair of lines represented by \[ 3x^2 - 5xy + P y^2 = 0 \] and \[ 6x^2 - xy - 5y^2 = 0 \] have one line in common, then the sum of all possible values of \( P \) is:
The area of the triangle formed by the lines represented by \( 3x + y + 15 = 0 \) and \( 3x^2 + 12xy - 13y^2 = 0 \) is:
For \( \lambda, \mu \in \mathbb{R} \), the lines \[ (x - 2y - 1) + \lambda (3x + 2y - 11) = 0 \] and \[ (3x + 4y - 11) + \mu (-x + 2y - 3) = 0 \] represent two families of lines. If the equation of the line common to both families is given by \[ ax + by - 5 = 0, \] then \( 2a + b = \) ?
Consider the tetrahedron with the vertices \( A(3,2,4) \), \( B(x_1,y_1,0) \), \( C(x_2,y_2,0) \), and \( D(x_3,y_3,0) \). If the triangle \( BCD \) is formed by the lines \( y = x \), \( x + y = 6 \), and \( y = 1 \), then the centroid of the tetrahedron is: