# Oxford PAT 2009, Questions 3 and 4

## 2 thoughts on “Oxford PAT 2009, Questions 3 and 4”

1. Leon Wise says:

Can you please explain how you got that tanß=3^0.5 and tan(ŷ)=-(3)^0.5, and also how that helps to find the equations. I can’t work out why the two tangents of the angles represent the gradients. Also, is there a way to do his with implicit differentiation?
Thanks very much.

1. Can you please explain how you got that tanß=3^0.5 and tan(ŷ)=-(3)^0.5

The triangle containing α is a right-angle triangle which has sides 4, 2, 2√3. So you should be able to see that it is a ’30, 60, 90′ triangle, and therefore α = π/6. So by using symmetry and similar triangles you should be able to show that ß=π/3, and γ=2π/3. Then by considering the properties of 30, 60, 90 triangles again you can see that tan ß = √3 and tan γ = -√3.

I can’t work out why the two tangents of the angles represent the gradients.

Have a look at this link which goes through the details of how the gradient of the line is equal to the tangent of the angle it makes with the horizontal: http://www.mathcentre.ac.uk/resources/uploaded/mc-ty-gradstlnseg-2009-1.pdf