Important points to remember
(i) The lines represented by ax
2
+ 2hxy + by
2
= 0 and
ax
2
+ 2hxy + by
2
+ 2gx + 2fy + c = 0 form:
(a) a parallelogram if a + b ≠ 0, (a – b)fg + h (f
2
– g
2
) ≠ 0
(b) a rectangle if a + b = 0, (a – b)fg + h (f
2
– g
2
) ≠ 0
(c) a rhombus if (a + b) ≠ 0, (a – b)fg + h (f
2
– g
2
) = 0
(d) a square if (a + b) = 0, (a – b)fg + h (f
2
– g
2
) = 0
(ii) The area of parallelogram formed by the two pairs of lines ax
2
+ 2hxy + by
2
= 0 and ax
2
+ 2hxy + by
2
+ 2gx + 2fy + c = 0 is given by
(iii) The product of the perpendicular distances from the origin to the pair of straight lines ax
2
+ 2hxy + by
2
+ 2gx + 2fy + c = 0 is given by
(iv) Length of the X-intercept made by the pair of lines ax
2
+ 2hxy + by
2
+ 2gx + 2fy + c = 0 is given by
(v) Length of the Y-intercept made by the pair of lines ax
2
+ 2hxy + by
2
+ 2gx + 2fy + c = 0 is given by
(vi) The product of the length of perpendiculars form (x
1
, y
1
) upon the lines ax
2
+ 2hxy + by
2
+ 2gx + 2fy + c = 0 is given by
(vii) The condition that the pair of lines represented by
ax
2
+ 2hxy + by
2
+ 2gx + 2fy + c = 0 intersect on X-axis is
g
2
= ac
(viii) The condition that the pair of lines represented by
ax
2
+ 2hxy + by
2
+ 2gx + 2fy + c = 0 intersect on Y-axis is
f
2
= bc
(ix) If ax
2
+ 2hxy + by
2
= 0 are two sides of a parallelogram and lx + my = 1 is one diagonal, then the equation to the other diagonal is
y(bl – hm) = x(am – hl)
(x) If (kl, km) is the orthocenter of the triangle formed by the lines ax
2
+ 2hxy + by
2
= 0 and lx + my = 1, then
k =
(xi) If ax
2
+ 2hxy + by
2
= 0 represents two sides of a triangle for which (c, d) is the orthocenter, then the equation of the third side is
(a + b) (cx + dy) = ad
2
– 2hcd + bc
2
(xii) The condition that the slopes of the lines represented by ax
2
+ 2hxy + by
2
= 0
are in the ratio p : q is ab given by
(p + q)
2
= 4h
2
pq