List of top Mathematics Questions on Family of Lines

If sum of reciprocals of intercepts of a line equals A.M. of \(\frac{2}{3}\) and \(\frac{4}{5}\), then point of concurrence is
The equations of the pairs of opposite sides of a parallelogram are \[ x^2-5x+6=0 \] and \[ y^2-6y+5=0. \] Then the equations of its diagonals are
If the straight lines $x + 2y - 9 = 0$, $3x + 5y - 5 = 0$ and $ax + by - 1 = 0$ are concurrent, then the straight line $22x - 35y = 1$ passes through the point:
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:
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 \( 2a + b = \) ?
The area of the triangle formed by the lines represented by \( 3x + y + 15 = 0 \) and \( 3x^2 + 12xy - 13y^2 = 0 \) is:

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: 

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:
Suppose the pairs of straight lines \(x^2 - 2axy - y^2 = 0\) and \(x^2 - 2bxy - y^2 = 0\) are such that each pair bisects the angles between the other two. Then \(ab =\)
The equations of the lines perpendicular to \[ x^2-5xy+4y^2=0 \] and passing through \((2,1)\) is:
If \[ ax^2-34xy-5y^2+2x+26y-5=0 \] represents a pair of straight lines, then the value of \(a\) is:
The equation of bisectors of the angle between the lines given by \[ 3x^2+5xy+4y^2=0 \] is:
The two pairs of straight lines \[ 12x^2+7xy-12y^2=0 \] and \[ 12x^2+7xy-12y^2-x+7y-1=0 \] constitute a
The family of lines, forming an isosceles triangle with the lines \(3x-4y-2=0\) and \(12x-5y+6=0\), is
The straight line passing through \((-1,1)\) and remaining parallel to the line common to the pairs of lines provided by \(6x^2-xy-12y^2=0\) and \(15x^2+14xy-8y^2=0\), is