Question

If the orthocentre of the triangle formed by the lines 2x + 3y – 1 = 0, x + 2y – 1 = 0 and ax + by – 1 = 0, is the centroid of another triangle, whose circumecentre and orthocentre respectively are (3, 4) and (–6, –8), then the value of |a– b| is_____.

Answer 1

Correct Answer: 16

\[ 2x + 3y = 1 \quad \text{(1)} \] \[ x + 2y = 1 \quad \text{(2)} \] \[ ax - by = 1 \quad \text{(3)} \] Points given: \( (6, -5), G(2, 6), (0, 0) \). Equation of a line: \[ x + 2y = 1 \] where \( x \) and \( y \) represent coordinates. Given equation: \[ ax - by = 1 \] with \( x = -1 \) and \( y = 0 \). From \[ ax - by = 1 \], substituting \( x = -1 \) and \( y = 0 \) yields \( -a = 1 \), so \( a = -1 \). Substituting \( a = -1 \) into \[ ax - by = 1 \) gives \[ -x - by = 1 \). If \( x = -1 \) and \( y = 0 \), then \[ -(-1) - b(0) = 1 \), which simplifies to \( 1 = 1 \). The condition \[ a = b \) implies \( a = -1 \) and \( b = -1 \). Therefore, the equation becomes \[ -x - (-1)y = 1 \Rightarrow -x + y = 1 \). This contradicts the derived condition \[ x(a - b) = 3 \), which simplifies to \( x(0) = 3 \Rightarrow 0 = 3 \) if \( a = b \). The provided information seems contradictory or incomplete. Assuming the final equation derived from the given points is intended to be \( 2x + 6y = 0 \) (by doubling equation (1) and setting RHS to 0), this does not align with the earlier equations. If \( x = 3 \), and considering the equation \[ ax - ay = 0 \), this implies \( a(3) - ay = 0 \). If \( a eq 0 \), then \( 3 - y = 0 \Rightarrow y = 3 \). If \( y = 6 \), and \( a = 8, b = 8 \), then \[ |a - b| = |8 - 8| = 0 \), not \( 16 \). There appear to be multiple inconsistencies in the provided data and derivations.