Straight lines

The line L has equation 3x − 4y + 12 = 0

1. Find the gradient of L.
2. Find the coordinates of the points where L crosses the x-axis and the y-axis.

The points A and B have coordinates (−2, 7) and (4, −2) respectively.

Find an equation of the line through A and B, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

The line l₁ has equation 2x − 5y + 3 = 0

The line l₂ is perpendicular to l₁ and passes through the point P(5, −1).

Find an equation of l₂ in the form y = mx + c

The points A(−3, 4) and B(5, 8) are joined to form the line segment AB.

Find an equation of the perpendicular bisector of AB, giving your answer in the form ax + by + c = 0, where a, b and c are integers.

The points A(1, k) and B(k, 9), where k is a constant, lie on a line that is perpendicular to the line with equation 3x + y = 7

Find the value of k.

The line l₁ has equation y = 3x − 5 and the line l₂ has equation x + 2y − 11 = 0

The lines meet at the point P, as shown in the sketch.

1. Find the coordinates of P.

The line l₁ crosses the x-axis at R and the line l₂ crosses the x-axis at Q.

2. Find the exact area of triangle PQR.

The points A(−1, 2), B(3, 5) and C(6, 1) are three vertices of the square ABCD.

1. Show that AB = BC.
2. Show that AB is perpendicular to BC.
3. Find the area of the square ABCD.
4. Find the coordinates of D.

The point A has coordinates (2, −1).

The point B lies on the line with equation y = 2x + 1 and the distance AB is 2√5

Find the possible coordinates of B.

The line l₁ has equation 4x + 3y = 24 and crosses the x-axis at A and the y-axis at B.

The line l₂ is parallel to l₁ and passes through the point C(9, 8).

1. Find an equation of l₂ in the form ax + by = c

The line l₂ crosses the x-axis at D.

2. Find the coordinates of D.
3. Explain why ABCD is a parallelogram and find its area.

A household's monthly gas bill, £C, is modelled by C = a + bE, where E is the number of kWh of gas used that month and a and b are constants.

In March the household used 820 kWh and the bill was £162.40. In June the household used 300 kWh and the bill was £68.80

1. Find the values of a and b.
2. Interpret the value of b in the context of the model.
3. Use the model to estimate the bill for a month in which 1500 kWh are used, and comment on the reliability of your estimate.

The triangle ABC has vertices A(−4, 1), B(6, −4) and C(2, 8).

1. Find an equation of the line through C that is perpendicular to AB.

This line meets AB at the point N.

2. Find the coordinates of N.
3. Hence find the area of triangle ABC. You must show all your working.

Two candles, A and B, are lit at the same time. Their heights are modelled as linear functions of the time t hours after lighting.

• Candle A is 30 cm tall when lit and 22 cm tall 4 hours later.
• Candle B is 24 cm tall when lit and burns down completely in 16 hours.

1. Find, for each candle, an equation for its height h cm in terms of t.
2. Find the time at which the two candles are the same height, and that height.
3. Find the interval of time for which candle A is still burning but is shorter than candle B.
4. Give one limitation of the model.