ADVANCED EXERCISE 09B: EQUATION OF STRAIGHT LINE
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1 ADVANCED EXERCISE 09B: EQUATION OF STRAIGHT LINE It is given that the straight line L passes through A(5, 5) and is perpendicular to the straight line L : x+ y 5= 0 (a) Find the equation of L (b) Find the coordinates of the intersection of L (c) Hence, find the perpendicular distance from A to L In the figure, A(5, 8), B(-, 3) and C(, -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 Page
2 3 In the figure, L is the horizontal line passing through A(0, -) L is the straight line passing through A and B(-, 4) L is the straight line passing through B and perpendicular to L C is the intersection of L (a) Find the equations of L, L (b) Find the coordinates of C (c) Find the area of ABC 4 In the figure, the two straight lines L : x+ y 9= 0 are perpendicular to each other, and intersect at P( a,5) A is a point on the x-axis such that PA is vertical L cuts the y-axis at B L is the straight line passing through A and B (a) (i) Find the value of a (ii) Find the equation of L (b) (i) Find the coordinates of A and B (ii) Find the equation of L (c) Explain whether L // L Page
3 5 In the figure, A is a point on the y-axis and B is a point on the x-axis P(6,) 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(, 0), B(, -5) and C(5, -) 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 Page 3
4 7 In the figure, L is the straight line passing through A(0, -) and B(6, 0) L is the straight line passing through C(0, 5) with slope - L intersect at D (a) Find the equations of L (b) (i) Show that L divides ABC into two triangles of equal area (ii) Hence, find the area of BCD 8 In the figure, A(-3, -5), B(5, ) and C(-3, 9) are the vertices of a triangle The straight lines L are the perpendicular bisectors of AC and BC in ABC respectively (a) Find the equations of L (b) Find the circumcentre of ABC Page 4
5 9 In the figure, O, P, Q(8, ) 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 y= 3x (a) Find the value of M (b) Find the equation of the straight line passing through (i) P and Q, (ii) R and Q 0 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 y= mx 4 ; the equation of BC is y= kx+ 6 (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 Page 5
6 In the figure, O, A(4, 8), B and C(c, 6) are the vertices of a trapezium, where OC // AB The equation of OC is x+ y+ k= 0 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 In the figure, A(9, a) is a point on the straight line L : hx 4y+ 3= 0 B is a point on the straight line L : 3hx 4y + k = 0 and 6 units vertically above A L cut the x-axis at the same point C D lies on AC (a) (i) Show that the slope of L is 3 times that of L (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 Page 6
7 3 In the figure, N and C lie on the x-axis, and M is a point on the line segment 5 joining A, 0 and B 3, 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 4 Given that the lines 4x+ y= 4, mx+ y= 0 and x 3my= 4 cannot form a triangle Suppose that m> 0 and Q is the minimum possible value of m, find Q Page 7
8 5 Suppose (, ) P a b is a point on the straight line x y+ = 0 such that the sum of the distance between P and the point A(, 0) and the distance between P and the point B(3, 0) is the least, find the value of a+ b 6 If the lines y= x+ d and x y d = + intersect at the point ( d, d), find the value of d Page 8
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