Geometry
# 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 \(ax + by + c = 0\) (with \(a,\) \(b,\) and \(c\) all coprime) 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.

\[ 6x^2-\alpha xy-3y^2-24x+3y+\beta=0 \]

The equation above represents a pair of lines that intersect on the \(x\)-axis. Find the value of \(20\alpha-\beta\).

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 the \(xy\)-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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