f.5 amaths (4 questions)

1)
Point of contact = (a, b)
Curve: y = x3 + 5
dy/dx = 3x2
Slope of the tangent = 3a2

3x - y = 0: Slope = -3/(-1) = 3
The tangent // 3x - y = 0
Slope of the tangent = 3

Hence, 3a2 = 3
a2 = 1
a2 - 1 = 0
(a - 1)(a + 1) = 0
a = 1 or a = -1

When a = 1 :
b = (1)3 + 5
b = 6

When a = -1
b = (-1)3 + 5
b = 4

Ans: The two points are (1, 6) and (-1, 4)

======
2)
Point of contact = (a, b)
Curve: y = 4/(1 - x)2
dy/dx = 8/(1 - x)3
Slope of tangent = 8/(1 - a)3

y = x : Slope = 1
The tangent // y = x
Slope of the tangent = 1

Hence, 8/(1 - a)3 = 1
(1 - a)3 = 8
1 - a = 2
a = -1

b = 4/(1 - a)2
b = 4/[1 - (-1)]2
b = 1

The point of contact = (-1, 1)
Slope of the tangent = 1
Point-slope form of the tangent:
y - 1 = 1(x + 1)
y - 1 = x + 1
Tangent: x - y + 2 = 0

======
3)
Point of contact = (a, b)
Curve: 4x2 - y2 = 15 ...... (1)
d/dx(4x2 - y2) = d/dx(15)
8x - 2y(dy/dx) = 0
2y(dy/dx) = 8x
dy/dx = 4x/y
Slope of the tangent = 4a/b

8x - y + 2 = 0 : Slope = -8/(-1) = 8
The tangent // 8x - y + 2 = 0
Slope of the tangent = 8

Hence, 4a/b = 8
a/b = 2
a = 2b ...... (2)

Put (2) into (1):
4(2b)2 - b2 = 15
15b2 = 15
b2 = 1
b = 1 or b = -1

When b = 1:
(2): a = 2(1) = 2
Slope = 8
Point-slope form of the tangent:
y - 1 = 8(x - 2)
y - 1 = 8x - 16
8x - y - 15 = 0

When b = -1:
(2): a = 2(-1) = 2
Slope = 8
Point-slope form of the tangent:
y + 1 = 8(x + 2)
y + 1 = 8x + 16
8x - y + 15 = 0

Tangent: 8x - y - 15 = 0 or 8x - y + 15 = 0

=====
4)a)
x2 +4y2 = 72
d/dx(x2 + 4y2) = d/dx(72)
2x + 8y(dy/dx) = 0
8y(dy/dx) = -2x
dy/dx = -x/(4y)

4)b)
C = (a, b)
a2 + 4b2 = 72 ...... (1)

dy/dx = -x/(4y)
Slope of the tangent at C = -a/(4b)

Slope of the tangent at C = (b - 4)/(a - 4)

Hence, (b - 4)/(a - 4) = = -a/(4b)
4b(b - 4) = -a(a - 4)
4b2 - 16b = -a2 + 4a
4a + 16b = a2 +4b2

But a2 + 4b2 = 72
Hence, 4a + 16b = 72
a + 4b = 18
a = 18 - 4b ...... (2)

Put (2) into (1):
(18 - 4b)2 + 4b2 = 72
324 - 144b + 16b2 + 4b2 = 72
20b2 - 144b + 252 = 0
5b2 - 36b + 63 = 0
(b - 3)(5b - 21) = 0
b = 3 or b = 21/5

When b = 3
a = 18 - 4(3)
a = 6

When b = 21/5
a = 18 - 4(21/5)
a = 6/5

Ans: C is (6, 3) or (6/5, 21/5)
=