 Geometry

# Equation of a Line: Level 4 Challenges

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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