Explore how simple equations have graphical implications starting with the equation of a line.

Given that a straight line in the \(xy\)-plane passes through the point \((3,48)\) in the first quadrant and intersects the positive \(x\)- and \(y\)-axes at points \(P\) and \(Q\) respectively. Let \(O\) designate the origin.

Find the minimum value of \(OP\times OQ\).

The line \(4x + y = 1\) passes through the point \(A(2,-7)\) and intersects point \(B\) on the line \(BC\) whose equation is \(3x - 4y + 1=0\). The equation of the line \(AC\) is \(ax + by + c = 0\) and constructed so that the distance from \(A\) to \(B\) is the same as the distance from \(A\) to \(C\).

What is the minimum positive value of \(a+b+c\)?

Note: \(A,B,C\) are all different points in the coordinate plane.

There exist two ordered triples \((a,b,c)\) for which \(4x^2-4xy+ay^2+bx+cy+1=0\) represents a pair of identical straight lines in \(x\)-\(y\) plane.

If these triples are \((a_1,b_1,c_1)\) and \((a_2,b_2,c_2)\), then find the value of \(a_1+b_1+c_1+a_2+b_2+c_2\).

Given that a straight line in the \(xy\)-plane passes through the point \((3,48)\) in the first quadrant and intersects positive \(x\)- and \(y\)-axes at points \(P\) and \(Q\) respectively. Let \(O\) designate the origin.

Find the minimum value of \(OP+OQ\).

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