1) Show that 𝑥 2 + 𝑥𝑦 − 6𝑦 2 − 𝑥 − 8𝑦 − 2 = 0 represent a line pair and calculate their angle of intersection.
2) Show that the line pair through the origin respectively perpendicular to the line pair 𝑎𝑥 2 + 2ℎ𝑥𝑦 + 𝑏𝑦 2 = 0 is given by 𝑏𝑥 2 − 2ℎ𝑥𝑦 + 𝑎𝑦 2 = 0.
3) The straight line 𝑙𝑥 + 𝑚𝑦 + 𝑛 = 0 cuts the distinct pair of straight lines
𝑎𝑥
2
+ 2ℎ𝑥𝑦 + 𝑏𝑦
2
= 0 at the points P and Q. If the angle OPQ equals the angle OQP,O being the origin. Show that ℎ(𝑙
2
− 𝑚2)= 𝑙𝑚(𝑎 − 𝑏).
4) Translate to the parallel axes through (4,-2) and hence simplify
16𝑥
2
+ 25𝑦
2
− 128𝑥 + 100𝑦 − 44 = 0.
5) Rotate axes for each of the following through the angle indicated.
I. 𝑥
2
+ 4𝑥𝑦 + 𝑦
2
= 2 ; 450
II. 𝑥𝑦 + 𝑥 + 𝑦 = 0 ; 450
III. 4𝑥
2
+ 4𝑦
2
= 17𝑥𝑦 ; 450
