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Mathematics St. Bonaventure College and High School
Form 4, Quiz 7 (Ch6: Equation of Straight Line – General Form & Possible intersection of straight line)
1
Name: Class: ( ) Date: Mark: /27
Time: 38min, Full Marks: 27 Section A (6 marks) (Working steps are NOT required in this section.) 1. If the two straight lines L1: kx + 2y + 5 = 0 and L2: 12x + 3y – 8 = 0
are parallel, find the value of k. ____________________
2. Determine the number of intersections between the two straight
lines L1: 3x – 2y – 8 = 0 and L2: 423 += xy . ____________________
3. It is given that the two straight lines L1: 2x – y = 1 and L2: 5x – 4y = 16 intersect at the point P. Find the coordinates of P. ____________________ 4. It is given that the two straight lines L1 and L2: y + 3 = 0 cuts the y-‐
axis at the same point. If L1 passes through (–9, 3), find the equation of L1. ____________________
Mathematics St. Bonaventure College and High School
Form 4, Quiz 7 (Ch6: Equation of Straight Line – General Form & Possible intersection of straight line)
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Section B (21 marks) 1. The figure shows two straight lines L1: x – y + 1 =
0 and L2. They intersect at A. The slope of L2 is the negative of the slope of L1, and the y-‐intercept of L2 is 5. Find (a) the equation of L2, (b) the coordinates of A. (5 marks)
Mathematics St. Bonaventure College and High School
Form 4, Quiz 7 (Ch6: Equation of Straight Line – General Form & Possible intersection of straight line)
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2. If two straight lines 𝐿!: 𝑘 + 1 𝑥 + 𝑘 − 1 𝑦 + 2 = 0 and 𝐿!: 3𝑥 − 4𝑦 + 5 = 0 are parallel, find the value of k. (4 marks)
Mathematics St. Bonaventure College and High School
Form 4, Quiz 7 (Ch6: Equation of Straight Line – General Form & Possible intersection of straight line)
4
3. In the figure, A and B are points on the y-‐axis and the x-‐axis respectively. The equation of AB is 5x-‐2y+10=0. (a) Find the area of triangle AOB. (4 marks) (bi) An ant travels from O towards the line segment AB. Using (a) or otherwise, find the shortest distance the ant travelled. (Give the answer in surd form if necessary.) (bii) Suppose the ant stops at C, which is a point on AB. If AC:BC=1:2, find the coordinates of C. (7marks)
A
B 0
5x-‐2y+10=0
Mathematics St. Bonaventure College and High School
Form 4, Quiz 7 (Ch6: Equation of Straight Line – General Form & Possible intersection of straight line)
5
Solution: Question Answers/Solutions Marks Remarks Section A
1 8 1 2 no intersections 1 3 (–4, –9) 2
4 332 −−= xy
or 2x+3y+9=0 2
Section B 1 (a)
1 of slope of Slope 12
−=−= LL
The equation of L2 is
∴
551
+−=+⋅−=
xyxy
(b)
)2(5:)1(01:
2
1
!!
!!
+−==+−
xyLyxL
By substituting (2) into (1), we have
204201)5(
==−=++−−
xx
xx
By substituting x = 2 into (2), we have
3
52=
+−=y
∴ Coordinates of A )3,2(=
1
1 1 1
1
Answer mark
Method mark Answer mark Method mark
Answer mark
Mathematics St. Bonaventure College and High School
Form 4, Quiz 7 (Ch6: Equation of Straight Line – General Form & Possible intersection of straight line)
6
2. 𝐿!: 𝑘 + 1 𝑥 + 𝑘 − 1 𝑦 + 2 = 0 𝐿!: 3𝑥 − 4𝑦 + 5 = 0 Slope of 𝐿! = − !!!
!!!= !!!
!!! …1A
Slope of 𝐿! = − !!!
= !! …1A
Therefore, we have Slope of L! = Slope of 𝐿! …1M
𝑘 + 11− 𝑘 =
34
4𝑘 + 4 = 3− 3𝑘 7𝑘 = −1
𝑘 = − !! …1A
3. (a)
𝑥 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑜𝑓 𝐴𝐵 = − !"!= −2,𝑦 − 𝑖𝑛𝑡𝑒𝑟𝑐𝑒𝑝𝑡 𝑜𝑓 𝐴𝐵 = !"
!!= 5 …1M+2A
𝐴𝑟𝑒𝑎 𝑜𝑓 ∆𝐴𝑂𝐵 = !"×!"!
= !!! [!! !! ]!
= 5 𝑠𝑞.𝑢𝑛𝑖𝑡𝑠 …1M+1A (bi) 𝐴𝐵 = 5! + 2! = 29 𝑢𝑛𝑖𝑡𝑠 𝑃𝑦𝑡ℎ.𝑇ℎ𝑒𝑜𝑟𝑒𝑚 …1M To travel with the shortest distance, the ant should reach a point D on AB such that OD is perpendicular to AB. Consider the area of triangle AOB: !!× 29×𝑂𝐷 = 5 …1M 𝑂𝐷 = !"
!" 𝑢𝑛𝑖𝑡𝑠 ….1A
𝑂𝐷 = !"!"× !"
!"= !" !"
!" 𝑢𝑛𝑖𝑡𝑠 …1A(Bonus, for Rationalize the denominator)
Therefore, the shortest distance is !" !"!"
units. (bii) Coordinates of A =(0,5) Coordinates of B = (-‐2,0) Coordinates of C = ! !! !! !
!!!, ! ! !! !
!!!= (− !
!, !"!) …1M+2A
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