Find the value of p for which the straight lines 8px + (2 - 3p)y + 1 = 0 and px + py - 7 = 0 are perpendicular to each other.

Find the equation of the parabola whose focus is the point F(-1, -2) and the directrix is the line 4x - 3y + 2 = 0

If the slope of one of the straight lines ax² + 2hxy + by² = 0 is thrice that of the other, then show that 3h² = 4ab.

If (4,1) is one extremity of a diameter of the circle x² + y² - 2x + 6y -15 = 0, find the other extremity.

The average variable cost of a monthly output of x tonnes of a firm producing a valuable metal is  Rs. \(\frac { 1 }{ 5 } { x }^{ 2 }-6x+100\). Show that the average variable cost curve is a parabola. Also find the output and the  average cost at the vertex of the parabola.