The straight lines x + 2y - 9 = 0,3x + 5y - 5 | vedclass

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Question

(A) The three lines are concurrent if the determinant of their coefficients is zero:$\left| \begin{array}{ccc} 1 & 2 & -9 \ 3 & 5 & -5 \ a & b & -1

Vedclass · Accepted Answer

$(a, b)$

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(A) The three lines are concurrent if the determinant of their coefficients is zero:$\left| \begin{array}{ccc} 1 & 2 & -9 \ 3 & 5 & -5 \ a & b & -1 \end{array} ight| = 0$Expanding the determinant along the third row:$a(2(-5) - 5(-9)) - b(1(-5) - 3(-9)) - 1(1(5) - 3(2)) = 0$$a(-10 + 45) - b(-5 + 27) - 1(5 - 6) = 0$$35a - 22b + 1 = 0$This equation implies that the point $(a, b)$ satisfies the equation of the line $35x - 22y + 1 = 0$.Therefore,the line passes through the point $(a, b)$.

The straight lines $x + 2y - 9 = 0$,$3x + 5y - 5 = 0$ and $ax + by - 1 = 0$ are concurrent,if the straight line $35x - 22y + 1 = 0$ passes through the point

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