Question Statement

Q2. Check whether the lines are concurrent or not:

(i) $3x-4y-13=0$ $8x-11y-33=0$ $2x-3y-7=0$

(ii) $x+2y-4=0$ $x-y-1=0$ $4x+5y-13=0$

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Background and Explanation

Three straight lines are concurrent if they all intersect at a single common point. For three lines given in the general form $ax+by+c=0$, the algebraic condition for concurrency is that the determinant of the $3\times 3$ coefficient matrix equals zero.

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Solution

Part (i)

The given lines are: $3x-4y-13=0,8x-11y-33=0,2x-3y-7=0$

Consider the determinant of the coefficients: $\left|\begin{array}{rrr}3& -4& -13\\ 8& -11& -33\\ 2& -3& -7\end{array}\right|$

Expanding along the first row: $\begin{array}{rl}& =3\left|\begin{array}{rr}-11& -33\\ -3& -7\end{array}\right|+4\left|\begin{array}{rr}8& -33\\ 2& -7\end{array}\right|-13\left|\begin{array}{rr}8& -11\\ 2& -3\end{array}\right|\\ & =3(77-99)+4(-56+66)-13(-24+22)\\ & =3(-22)+4(10)-13(-2)\\ & =-66+40+26\\ & =0\end{array}$

Thus, the given lines are not concurrent.

Part (ii)

The given lines are: $x+2y-4=0,x-y-1=0,4x+5y-13=0$

Consider the determinant: $\begin{array}{rl}& \left|\begin{array}{rrr}1& 2& -4\\ 1& -1& -1\\ 4& 5& -13\end{array}\right|\\ & =1\left|\begin{array}{rr}-1& -1\\ 5& -13\end{array}\right|-2\left|\begin{array}{rr}1& -1\\ 4& -13\end{array}\right|-4\left|\begin{array}{rr}1& -1\\ 4& 5\end{array}\right|\\ & =1(13+5)-2(-13+4)-4(5+4)\\ & =18+18-36\\ & =0\end{array}$

Since the value of the determinant is zero, the given lines are concurrent.

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Key Formulas or Methods Used

• Condition for concurrency: Three lines $a_{1}x+b_{1}y+c_1=0$, $a_{2}x+b_{2}y+c_2=0$, and $a_{3}x+b_{3}y+c_3=0$ are concurrent if and only if: $\left|\begin{array}{ccc}{a}_1& b_1& c_1\\ a_2& b_2& c_2\\ a_3& b_3& c_3\end{array}\right|=0$

• Determinant expansion: Expansion of a $3\times 3$ determinant using cofactors along any row or column, where each element is multiplied by its corresponding minor with appropriate sign $(-1{)}^{i+j}$

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Summary of Steps

1. Identify the coefficients $(a,b,c)$ from each line equation $ax+by+c=0$
2. Construct the $3\times 3$ determinant with rows formed by the coefficients of each line
3. Expand the determinant along the first row (or any convenient row/column) using the cofactor method
4. Calculate the $2\times 2$ minors by finding the difference of cross-products
5. Simplify the expression: if the final value equals zero, the lines are concurrent; otherwise, they are not concurrent