C1 Help

Can anyone help me?

A curve has the equation y = x/2 + 3 - 1/x x isn't equal to 0

The point A on the curve has x - coordinate 2.

(i) Find the gradient of the curve at A.

(ii) Show that the tangent to the curve at A has equation

3x - 4y + 8 = 0

The tangent to the curve at the point B is parallel to the tangent at A.

(iii) Find the coordinates of B.

The straight line l has gradient 3 and passes through point A (-6,4)

(i) Find an equation for l in the form y = mx + c.

The straight line m has the equation x - 7y + 14 = 0

Given that m crosses the y axis at the point B and intersects l at the point C,

(ii) Find the coordinates of B and C,

(iii) Show that angle BAC = 90,

(iv) Find the area of triangle ABC.

Thanks =)

Reply 1

Apagg

I'll do it one step at a time, as I know if do the whole thing someone will enter their answer just before mine
i)
To find the gradient, take dy/dx

1/2 + 1/x^2

We know that x = 2
So 1/2 + 1/4
=3/4

Gradient = 3/4

Reply 2

Apagg

ii) At A, x=2, so sub into the curve equuation

y = 2/2 +3 -1/2
y= 7/2

The equation of a line is

y = mx + c
So
7/2 = 3/4 * 2 + c
14/4 - 6/4 = c
c= 2

So equation of the tangent is

y = 3/4x + 2
(*4) 4y = 3x + 8
3x - 4y + 8 = 0

Reply 3

Apagg

Sorry for the delay

iii) Parallel, so dy/dx = 3/4

Therefore 1/2 - 1/x^2 = 3/4
1/x^2 = 1/4
x = +/- 2

As x = 2 for A, it must = -2 for B.

Sub this into the orginal equation and you get
y = -1 + 3 + 1/2
= 5/2

So B (-2, 5/2)

Reply 4

Apagg

Question 2,

i)
Gradient 3, y = -6, x = 4, so

-6 = 3*4 + c
c = -6 - 12
c = -18

y= 3x -18

Reply 5

Apagg

ii) At point B, x = 0 as the line crosses the y axis.
As x = 0,
7y = x + 14
7y = 14
y = 2

As for C

If it intersects l, you can combine the two equations, so

y = 3x - 18 [1]
7y = x + 14 [2]

Divide [2] by 7

y = x/7 + 2
Now

x/7 + 2 = 3x -18
x + 14 = 21x - 126
20x = 140
x =7

Therefore
y = 21-18
y = 3

B (0, 2) C (7, 3)

Reply 6

Apagg

Part iii) Is difficult to do here, but simply find AB, BC, CA, and then show that
h^2 = a^2 + b^2 is true.
The area can be found by the same principle using the appropriate formula