急F4 Equations of Straight Line

26.
(a)
Slope of AC = (-1 + 2) / (1 + 3) = 1/4
BD⊥AC and they bisect each other.
Slope of BD = -1 / (1/4) = -4
Mid-pt. of BD = Mid-pt of AC = ((-3+1)/2, (-2-1)/2) = (-1, -3/2)

Equation of BD (point-slope form) :
y + (3/2) = -4(x + 1)
2y + 3 = -8x - 8
8x + 2y + 11 = 0

(b)
Let (a, 0) be the coordinate of B.
Put x = a and y = 0 into the equation of BD:
8a + 0 + 11 = 0
a = -11/8
Hence, B = (-11/8, 0)

Let (b, c) be the coordinate of D.
[b + (-11/8)]/2 = -1 and (c + 0)/2 = -3/2
b = -5/8 and c = -3
Hence, D = (-5/8, -3)


27.
(a)
When l1 and l2 have infinitely many points ofintersection, l1 and l2 and coincide.
m/7 = -3/5 and -3/5 = 11/(-n)
5m = -21 and 3n = 55
m = -21/5 and n = 55/3

(b)
When l1 and l2 have no points of intersection, l1// l2 but they are not coincide.
m/7 = -3/5
m = -21/5
and n can be any real number, but n ≠ 55/3

(c)
When l2 and l2 have one point of intersection, l1and l2 are neither // nor coincide.
m can be any real number, but m ≠ -21/5
and n can be any real number.


28.
(a)
AM = BM (given)
∠AMP = ∠BMP = rt.∠ (given)
PM = PM (common side)
ΔAMP ≡ ΔBMP (SAS)
AP = BP (corr. sides of cong. Δs)

(b)
(i)
Slope of AB = (4 + 4)/ (-4 - 4) = -1
L⊥AB. Hence, slope of L = -1/(-1) = 1
L passes (0, 0)
Equation of L (point-slope form):
y - 0 = 1(x - 0)
x - y = 0

(ii)
P(r, s) lies on L. Put x = r and y = s into the equation of L:
r - s = 0
r = s

AP
= √[(r - 4)² + (s + 4)²]
= √[(s - 4)² + (s + 4)²]
= √(s² - 8s + 16 + s² + 8s + 16)
= √(2s² + 32)

BP
= √[(r + 4)² + (s - 4)²]
= √[(s + 4)² + (s - 4)²]
= √(s² + 8s + 16 + s² - 8s + 16)
= √(2s² + 32)

Hence, AP = BP