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Secondary 3 Mathematics Advanced Exercise 09b: Equation of
Straight Line
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ADVANCED EXERCISE 09B: EQUATION OF STRAIGHT LINE
1. It is given that the straight line 1L passes through A(5, 5)
and is perpendicular to the straight line 2 : 2 5 0L x y+ − = .
(a) Find the equation of 1L .
(b) Find the coordinates of the intersection of 1L and 2L .
(c) Hence, find the perpendicular distance from A to 2L .
2. In the figure, A(5, 8), B(-2, 3) and C(2, -3) are the
vertices of a triangle. M and N are
the mid-points of AB and AC respectively. P is the mid-point of
MN. (a) Find the coordinates of M, N and P. (b) If L is the
straight line passing through A and P, show that L also passes
through
the mid-point of BC.
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Secondary 3 Mathematics Advanced Exercise 09b: Equation of
Straight Line
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3. In the figure, L is the horizontal line passing through A(10,
-2). 1L is the
straight line passing through A and B(-2, 4). 2
L is the straight line passing
through B and perpendicular to 1L . C is the intersection of L
and 2L .
(a) Find the equations of L, 1L and 2L .
(b) Find the coordinates of C. (c) Find the area of ABC∆ .
4. In the figure, the two straight lines : 2 9 0L x y+ − = and
1L are perpendicular to each
other, and intersect at ( ),5P a . A is a point on the x-axis
such that PA is vertical. 1L
cuts the y-axis at B. 2L is the straight line passing through A
and B.
(a) (i) Find the value of a.
(ii) Find the equation of 1L .
(b) (i) Find the coordinates of A and B.
(ii) Find the equation of 2L .
(c) Explain whether 2L // L.
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Secondary 3 Mathematics Advanced Exercise 09b: Equation of
Straight Line
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5. In the figure, A is a point on the y-axis and B is a point on
the x-axis. P(6,2) is a point on the coordinate plane such that PA
= OA and AP PB⊥ .
(a) Find the coordinates of A. (b) Find the equation of the
straight line passing through P and B. (c) (i) Prove that OAB PAB∆
≅∆ .
(ii) Hence, or otherwise, find the area of the quadrilateral
OAPB.
6. In the figure, A(1, 0), B(2, -5) and C(5, -2) are the
vertices of a triangle. AP is the altitude of BC in ABC∆ .
(a) Find the equations of BC and AP. (b) Find the length of
AP.
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Secondary 3 Mathematics Advanced Exercise 09b: Equation of
Straight Line
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7. In the figure, 1L is the straight line passing through A(0,
-2) and B(6, 0). 2L is the
straight line passing through C(0, 5) with slope -2. 1L and 2L
intersect at D.
(a) Find the equations of 1L and 2L .
(b) (i) Show that 2L divides ABC∆ into two triangles of equal
area.
(ii) Hence, find the area of BCD∆ .
8. In the figure, A(-3, -5), B(5, 1) and C(-3, 9) are the
vertices of a triangle. The straight lines
1L and 2L are the perpendicular bisectors of AC and BC in ABC∆
respectively.
(a) Find the equations of 1L and 2L .
(b) Find the circumcentre of ABC∆ .
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Secondary 3 Mathematics Advanced Exercise 09b: Equation of
Straight Line
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9. In the figure, O, P, Q(8, 2) and R are the vertices of a
rectangle. The equation of the straight line passing through O and
P is y mx= ; the equation of the straight line passing through O
and R is 3y x= .
(a) Find the value of M. (b) Find the equation of the straight
line passing through
(i) P and Q, (ii) R and Q.
10. In the figure, ABCD is a parallelogram. The coordinates of A
are (6, 8). B and D are points on the x-axis. The
equation of AB is 4y mx= − ; the equation of BC is 6y kx= +
.
(a) (i) Find the value of m. (ii) Find the coordinates of B.
(b) Find the value of k. (c) (i) Find the equation of AD.
(ii) Find the coordinates of D. (d) Find the area of the
parallelogram ABCD.
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Secondary 3 Mathematics Advanced Exercise 09b: Equation of
Straight Line
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11. In the figure, O, A(14, 8), B and C(c, 6) are the vertices
of a trapezium, where OC // AB. The equation of OC is 2 0x y k+ + =
. AB cuts the y-axis at P. BC
is parallel to the y-axis. (a) Find the values of c and k. (b)
Find the equation of AB. (c) Find the coordinates of B and P. (d)
Find the area of the trapezium OABC.
12. In the figure, A(9, a) is a point on the straight line 1 : 4
3 0L hx y− + = . B is a
point on the straight line 2 : 3 4 0L hx y k− + = and 6 units
vertically above A.
1L and 2L cut the x-axis at the same point C. D lies on AC.
(a) (i) Show that the slope of 2L is 3 times that of 1L .
(ii) Hence, find the value of a. (b) Find the values of h and k.
(c) Find the coordinates of C. (d) If the area of BCD∆ is 3 times
that of ABD∆ , find the equation of BD.
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Secondary 3 Mathematics Advanced Exercise 09b: Equation of
Straight Line
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13. In the figure, N and C lie on the x-axis, and M is a point
on the line segment
joining 5
, 02
A −
and 13
,122
B
. CM is the perpendicular bisector of AB. BN
and CM intersect at K, and BN : CM = 6 : 5. (a) (i) Find the
coordinates of M.
(ii) Find the equation of CM. (b) Find the coordinates of C, N
and K. (c) Find the ratio area of BKC∆ : area of KNC∆ . 14. Given
that the lines 4 4x y+ = , 0mx y+ = and 2 3 4x my− = cannot form a
triangle. Suppose that 0m> and Q
is the minimum possible value of m, find Q.
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Secondary 3 Mathematics Advanced Exercise 09b: Equation of
Straight Line
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15. Suppose ( ),P a b is a point on the straight line 1 0x y− +
= such that the sum of the distance between P and the
point A(1, 0) and the distance between P and the point B(3, 0)
is the least, find the value of a b+ .
16. If the lines y x d= + and x y d=− + intersect at the point (
)1,d d− , find the value of d.
