The Number Of Straight Lines Equidistant From Three Non-collinear Points In The Plane Of The Points Is Equal To

The sine of the angle between the pair of lines represented by the equation x² – 7xy + 12y² = 0 is 

The square of the differences of the slopes of the lines represented by the equation x²(sec²θ – sin²θ) – (2xy tan θ + y² sin²θ = 0) is 

The line joining the origin to the points of intersection of x² + y² + 2gx + c = 0 and x² + y² + 2fy – c = 0 are at right angles, if

Two of the lines represented by x³ – 6x²y + 3xy² + dy³ = 0 are perpendicular for   

If pairs of lines 3x² – 2pxy – 3y² = 0 and 5x² – 2qxy – 5y² = 0 are such that each pair bisects the angle between the other pair, then pq is equal to

The line x + y = 1 meets the line represented by the equation y³ –xy² – 14x²y + 24x³ = 0 at the points A, B, C. If O is the origin, then

OA² + OB² + OC² is equal to    

If the area of the triangle formed by the pair of lines 8x² – 6xy + y² = 0 and the line 2x + 3y = a is 7, then a is equal to

If p is length of the perpendicular from the origin on the line  are in A.P. then ab is equal to  

If two of the lines given by the equation ax³ – 9yx² – y²x + 4y³ = 0 are perpendicular then a is equal to

A line bisecting the ordinate PN of a point P(at², 2at), t > 0 on the parabola y2 = 4ax, a > 0 is drawn parallel to the axis to meet the curve at Q. If NQ and the tangent at the point P meet at, T, then the coordinates of point T is (where point N lie on the axis)