Determine if there exists any pair of linear equations where one line passes through (a, b) and (c, d) while the other passes through (b, a) and (d, c), such that they are parallel
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Step-by-Step Solution
Step 1: Find the slope of the first line
The slope of a line passing through two points (x1,y1) and (x2,y2) is given by the formula m=x2−x1y2−y1. For the first line passing through (a, b) and (c, d), we use this formula to find its slope, m1.
Step 2: Find the slope of the second line
Similarly, for the second line passing through (b, a) and (d, c), we apply the same slope formula. This gives us the slope m2=d−bc−a.
Step 3: Apply the condition for parallel lines
For two lines to be parallel, their slopes must be equal. Therefore, we set m1 equal to m2. This gives us the equation c−ad−b=d−bc−a.
Step 4: Solve the equation for the condition
To solve the equation, we cross-multiply, which results in (d−b)2=(c−a)2. Taking the square root of both sides, we get d−b=±(c−a). This implies two possible conditions: d−b=c−a or d−b=−(c−a).
Step 5: Analyze the conditions
From the first condition, d−b=c−a, we can rearrange the terms to get d+a=c+b. From the second condition, d−b=−(c−a), we get d−b=−c+a, which rearranges to d+c=a+b. If either of these conditions holds, the lines will be parallel.