Q1

The line CD contains points with coordinates (-3, 10), (0, 4) and (k, -7). Do you know why there is (0, 4)?

Gradient of CD
= Gradient of line joining (-3, 10) and (0, 4)
= (4 - 10) / [0 - (-3)]
= -6 / 3
= -2

All three points are on the same straight line, so the gradient of the line joining (0, 4) and (k, -7) must also be -2.

Gradient of line joining (0, 4) and (k, -7)
= (-7 - 4) / (k - 0)
= -11 / k

So, -11 / k = -2
-11 = -2k
2k = 11
k = 5.5

For the third part, the question involves points (5.5, -7) and (0, -9).

Gradient of this line
= [-9 - (-7)] / (0 - 5.5)
= (-2) / (-5.5)
= 2 / 5.5
= 4 / 11 (multiply 2 to both numerator and denominator)

So, the equation of this line takes the form y = 4/11 times x + c.

Since (0, -9) is on this line,

-9 = -4/11 (0) + c
c = -9

So, the equation of this line is y = (4/11) x - 9.

Q2

Length of PQ
= sqrt { [3 - (-2)]² + (-6 - 15)² }
= sqrt {5² + (-21)²}
= sqrt 466
= 21.58703314 units
~ 21.6 units

Gradient of PQ
= Gradient of line joining (-2, 15) and (3, -6)
= (-6 - 15) / [3 - (-2)]
= -21 / 5
= -4.2

So, the equation of PQ takes the form y = -4.2x + c.

Since (3, -6) is on this line,

-6 = -4.2 (3) + c
-6 = -12.6 + c
6.6 = c

So, the equation of PQ is y = -4.2x + 6.6.

Where line PQ crosses the x-axis,
y = 0
-4.2x + 6.6 = 0
-4.2x = -6.6
4.2x = 6.6
42x = 66
x = 11/7

So, the coordinates of this point is (11/7, 0).

If the point (2, -9) is to lie on PQ, its x and y coordinates must satisfy the equation of the line PQ nicely.

When x = 2,
y = -4.2 (2) + 6.6
y = -8.4 + 6.6
y = -1.8

This is obviously not the same as y = -9.

So, the point (2, -9) does not lie on line PQ.

Q3

Gradient of PQ
= (-3 - 1) / (-4 - 4)
= (-4) / (-8)
= 1/2

So, the equation of PQ takes the form y = 0.5x + c.

Since (4, 1) is on PQ,
1 = 0.5 (4) + c
1 = 2 + c
c = -1

So, the equation of PQ is y = 0.5x - 1.

If (k, -2.5) lies on PQ,
-2.5 = 0.5k - 1
-1.5 = 0.5k
-3 = k

Point P is (4, 1).

The line y + 2x = 6 can be rearranged into y = -2x + 6 with gradient -2, so the gradient of the line which we wish to form also has gradient -2.

y = -2x + c

Since (4, 1) lies on this line,
1 = -2 (4) + c
c = 9

So, the equation of this line is y = -2x + 9.

Thank u so much!!