Straight Lines and Triangle Centres: questions with solutions
3 free questions from this chapter (the full set has 30). Marking: Multi-correct: +4 all correct, +1 per correct option if none wrong, −2 if any wrong · Numerical: +4 / 0 · Matching: +3 / −1.
Question 1Level 2 · ExamSingle correct
The lines y=0, 12x−5y=0 and 4x+3y=56 form a triangle T. Of the four points that are equidistant from all three lines, let P be the one with negative y-coordinate, and let I be the incentre of T. The distance PI is ______. Exactly one option is correct.
A413
B265
C85
D16
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Answer: B
Vertices: y=0 and 12x−5y=0 meet at B(0,0); y=0 and 4x+3y=56 meet at C(14,0); 12x−5y=0 and 4x+3y=56 meet at A(5,12). Side lengths: a=BC=14, b=CA=81+144=15, c=AB=13.
The points equidistant from the three side lines are the incentre and the three excentres.
I=a+b+caA+bB+cC=42(70+182,168)=(6,4).
IA=−a+b+c−aA+bB+cC=14(−70+182,−168)=(8,−12), IB=a−b+caA−bB+cC=(21,14), IC=a+b−caA+bB−cC=(−7,221).
Only IA (which lies beyond side BC, on the side of y=0 away from A) has negative y-coordinate, so P=(8,−12). Check: its distances from the three lines are 12, 13∣96+60∣=12, 5∣32−36−56∣=12.
PI=22+162=260=265. Answer (B).
Difficulty: Hard
Question 2Level 2 · ExamSingle correct
The set of all real k for which the lines x+ky=1, kx+y=1 and x+y=k are the three side lines of a non-degenerate triangle is ______. Exactly one option is correct.
AR∖{−2,1}
BR∖{−1,1}
CR∖{−2,−1,1}
DR∖{−1,1,2}
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Answer: C
Three lines form a triangle exactly when no two are parallel (or coincident) and the three are not concurrent.
Parallel pairs: x+ky=1 and kx+y=1 are parallel iff 1⋅1−k⋅k=0, i.e. k=±1; x+ky=1 and x+y=k iff k=1; kx+y=1 and x+y=k iff k=1.
For k=1 all three lines are x+y=1. For k=−1 the lines x−y=1 and −x+y=1 are parallel and distinct, so there is no triangle.
Concurrency (for pairwise non-parallel lines): 1k1k11−1−1−k=(1−k)−k(1−k2)−(k−1)=(1−k)(2−k−k2)=(1−k)2(k+2).
So for k=−2 the lines are concurrent: x−2y=1, −2x+y=1, x+y=−2 all pass through (−1,−1).
For every other k the lines are pairwise non-parallel and not concurrent, so they form a triangle. The set is R∖{−2,−1,1}. Answer (C).
Difficulty: Hard
Question 3Level 2 · ExamSingle correct
For a real number m, let θ(m)∈[0,2π] denote the angle between the lines y=mx and y=2x. The set of all real m with θ(m)<4π is ______. Exactly one option is correct.
A(−∞,−3)∪(31,∞)
B(31,3)
C(−3,31)
D(−∞,−31)∪(3,∞)
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Answer: A
If 1+2m=0, i.e. m=−21, the lines are perpendicular and θ=2π, which is excluded.
Otherwise tanθ=1+2mm−2, and θ<4π iff tanθ<1 iff (m−2)2<(1+2m)2.
This is m2−4m+4<4m2+4m+1, i.e. 3m2+8m−3>0, i.e. (3m−1)(m+3)>0, so m<−3 or m>31. The value m=−21 is in neither part, as required.
So the set is (−∞,−3)∪(31,∞). For example m=−10 works: tanθ=1912<1. Answer (A).
Difficulty: Hard
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