F.4MATHS(3)

a)since x+2y+k=0 pass through (0,0) : 0+2*0+k=0 , k = 0
c is a point on x+2y=0 :
c+2(6)=0
c = -12
b)AB//OC :
the slope of OC x+2y=0 ie y = (-1/2)x ,slope = -1/2
The equation of AB is (y-8)/(x-14) = -1/2
2(y-8) = 14-x
2y - 16 = 14 - x
x + 2y - 30 = 0
c)Since BC//the y-axis and passing through AB, let B be (c , yb) = (- 12 , yb)
-12 + 2(yb) - 30 = 0
yb =21
B is (-12 , 21)
Let P be (0 , yp) :
0 + 2(yp) - 30 = 0
yp = 15
P is (0 , 15)
d)Let E be a point on AB such that OE丄AB , OE is the height of OABC,
The slope of OE = (- 1 / the slope of OC) = - 1 / (-1/2) = 2
The equation of OE is (y-0) / (x-0) = 2 ,ie y = 2x
slove y = 2x and AB equation x+2y-30=0 : x=6 , y=12
we get E is (6 , 12)
OE = √(6^2 + 12^2) = 6√5
OC = √(6^2 + (-12)^2) = 6√5
AB = √[ (14 + 12)^2 + (8 - 21)^2] = 13√5
OABC = (AB + OC)*OE/2
= (13√5 + 6√5)* 3√5
= 285