i might need more help with the next parts when I get to it but I'm going to get on with what I can do, thanks!
The angle between two circles at a point where they intersect is equal to the angle between the tangents to the circles at that point. So you need to find the slopes of the tangents to the two circles at B.
A tangent to a circle is perpendicular to the radius at that point, so we'll start by finding the slopes of the radii at B.
For the smaller circle, its centre is at the point (1,0). And B is the point (0,√3). The radius is the line joining (1,0) and (0,√3), and its slope is –√3. The tangent is perpendicular to the radius, so its slope is 1/√3. This means that the tangent makes an angle with the x-axis.
For the larger circle, its centre is at the point (3,0). The radius in this case is the line joining (3,0) and (0,√3), and its slope is –√3/3 = –1/√3. The tangent is perpendicular to the radius, so its slope is √3. This means that the tangent makes an angle with the x-axis.
Therefore the angle between the two tangents is radians. Finally, and so
do you need to do a differentiation on the circle equations? to find the angle of the tangents.
how does the SLOPE (GRADIENT?)help you to find the ANGLES? i can work out the two tangers
i understand what you are doing subtracting the smaller angle from the bigger angle using the xaxis for both to isolate the x angle. but i still dont know where your tan expressions are coming from. also is there a way of finding x straight away without doing the subtracting? (just so i know)
also does a tangent have just a slope or is it a line equation that i need? do i need to do the m formula?
(are u just using that the slopes forms triangles with the xaxis and using trig
can you not just use the slopes of the radiuses themselves instead of the tangents and use that the opposite angle is equal. because you can make the triangles with the x axis and the radiuses to B and use the triangle with the y axis to isolate the angle with the two circle centres atB, it is equal to x?)
i can work out what to do with te angle at c because it isnt like you perscribed with the two cricles. it is circle and one vertical line so i dont know what to do. i mean surely its differrent right. the hyperbolic angle being the same with the two cricles is because they both fold away form each toher? so with a circle and a line its not going to be the same...=/
i really can't figure out what to do at point C...
with B you have the two circles and you can use the TWO tangets to find the answer. C is made with a circle and a straight line
opalg on that last point
i thought the angle between the tangets coudl be used because the two circles 'bend away' from each other thus taking the tangets either side makes the angle equal to the angle with the two tangets?
do you not need to compensate for the fact thatyoure looking for the angle with a line one one side and a circle on the other? such that it wouldnt be equal to the angle between the line and the tanget?
but using this method i make it -1/root 3? what do you think? am i doing it right
or you can see that the tangent at B and C are equal just one negative and one positive cos they are oppositie again... just either side of the circle l
its backwards so the angle is negative? do we just make it positive?>?
if you do it backwards pi/2 - (- pi/6), if you do it forwards, pi/2 - pi/6? different results. can you have a negative angle. it doesnt seem right. i made the anser to y = pi/3. and then just take cos and sin for that as the answers. but the answers are in numbers not expressions. is that alright is that ok.
nilding who r u
edit please now help me with part 4. i have found two formulas... oen usus integrals, one does not use integrals. am i supposed to be looking at one with integrals or one without. for the hyperbolic arc length...
is the length of a, arccosh(5/3), can someone verify, and is the ln form, ln(3). just ln(3)? as the answee?
i am only 14 why am i doing this my mates are dont even know what a quadratic is
edit also as for AC, is it not just root 11 - root 3? its just a line, you can get the length by reading it off the coordinates just like that...?
this other thread) who are all working on this problem. I suggest that you should get in touch with each other. You can use the private message feature in this forum to do that. In the long run, you will find it more useful to share ideas with each other than to keep coming for help here.