If (-4, 7) is one vertex of a rhombus and if the equation of one diagonal is 5x - y + 7 = 0, then find the equation of another diagonal.

Find the locus of a point P that moves at a constant distant of 
(i) two units from the X-axis
(ii) three units from the Y-axis

Find the value of k and b, if the points P(-3, 1) and Q(2, b) lie on the locus of x² - 5x + ky = 0.

If O is origin and R is a variable point on y² = 4x, then find the equation of the locus of the mid-point of the line segment OR.

Find the equation of the lines passing through the point of intersection lines 4x - y + 3 = 0 and 5x + 2y + 7 = 0
(i) through the point (-1, 2)
(ii) Parallel to x - y + 5 = 0
(iii) Perpendicular to x - 2y + 1 = 0.

Find the equations of two straight lines which are parallel to the line 12x + 5y + 2 = 0 and at a unit distance from the point (1, -1).

If P is length of perpendicular from origin to the line whose intercepts on the axes are a and b, then show that \(\frac{1}{p^2}=\frac{1}{a^2}+\frac{1}{b^2}\)

Show that 2x² + 3xy - 2y² + 3x + y + 1 = 0 represents a pair of perpendicular lines.

Find the equation of the line, if the perpendicular drawn from the origin makes an angle 30° with x-axis and its length is 12

Find the value of λ for which the equation 12x²-10xy+2y²+11x-5y+λ=0 represents a pair of straight lines.