Geometry

Equation of a Line

Equation of a Line: Level 4 Challenges

         

Given that a straight line in the xyxy-plane passes through the point (3,48)(3,48) in the first quadrant and intersects the positive xx- and yy-axes at points PP and QQ respectively. Let OO designate the origin.

Find the minimum value of OP×OQOP\times OQ.

The line 4x+y=14x + y = 1 passes through the point A(2,7)A(2,-7) and intersects point BB on the line BCBC whose equation is 3x4y+1=03x - 4y + 1=0. The equation of the line ACAC is ax+by+c=0ax + by + c = 0 (with a,a, b,b, and cc all coprime) and constructed so that the distance from AA to BB is the same as the distance from AA to CC.

What is the minimum positive value of a+b+ca+b+c?

Note: A,B,CA,B,C are all different points in the coordinate plane.

6x2αxy3y224x+3y+β=0 6x^2-\alpha xy-3y^2-24x+3y+\beta=0

The equation above represents a pair of lines that intersect on the xx-axis. Find the value of 20αβ20\alpha-\beta.

There exist two ordered triples (a,b,c)(a,b,c) for which 4x24xy+ay2+bx+cy+1=04x^2-4xy+ay^2+bx+cy+1=0 represents a pair of identical straight lines in the xyxy-plane.

If these triples are (a1,b1,c1)(a_1,b_1,c_1) and (a2,b2,c2)(a_2,b_2,c_2), then find the value of a1+b1+c1+a2+b2+c2a_1+b_1+c_1+a_2+b_2+c_2.

Given that a straight line in the xyxy-plane passes through the point (3,48)(3,48) in the first quadrant and intersects positive xx- and yy-axes at points PP and QQ respectively. Let OO designate the origin.

Find the minimum value of OP+OQOP+OQ.

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