The sum of the squares of the distances of a moving point from two fixed points (a, 0) and (-0, 0) is equal to 2c². Find the equation to its locus.

Determine x so that the line passing through (3, 4) and (x, 5) makes 135° with the positive direction of x-axis.

Find the values of k for which the line (k - 3)x-(4-k²)y+(k²-7k + 6) = 0 passes through the origin.

Two sides of a square lie on the lines x + y = 1 and x + y + 2 = 0. What is its area?

If 9x² + 12xy + 4y² + 6x + 4y - 3 = 0 represents two parallel lines, find the distance between them.

If the line y = mx is one of the bisectors of the lines x² + 4xy - y² = 0, then find m.

Find the acute angle between the pair of lines given by 2x²- 5xy - 7y² = 0.

Find the equation of straight line joining the points of intersection of the lines 3x + 2y + 1 = 0 and x + y = 3 to the intersection of the lines y - x = 1 and 2x + y +2 = 0.

Find the equation of the straight line through the intersection of 5x - 6y = 1 and 3x + 2y + 5 = 0 and perpendicular to the straight line 3x - 5y + 11 =0.

Find the combined equation of the straight lines through the origin one of which is parallel to and the other is perpendicular to the straight line 3x + y + 5 = 0.

If the equation 12x² - 10xy + 2y² + 14x - 5y + k = 0 represents a pair of straight lines, find k, find separate equation and also angle between them.

Find the combined equation of the straight lines whose separate equations are x - 2y - 3 = 0 and x + y + 5 = 0.

A line passing through the points (a, 2a) and (-2, 3) is perpendicular to the line 4x+3y+ 5 = 0, find the value of a.

Find the angle between the pair of straight lines given by
(a² - 3b²)x² + 8ab xy+(b² -3a²)y² =0.