The rectangular hyperbola H has equation xy=c2, where c is a constant. The point P(ct,tc) is a general point on H.
a) Show that the tangent to H at P has equation
t2y+x=2ct
The tangents to H at the points A and B meet at the point (15c,−c)
b) Find, in terms of c, the coordinates of A and B
I'm only struggling with b)
The solution subs the x & y coordinate of the point (15c, -c) into t2y+x=2ct and cancels all the c's and has an equation for t and magically finds the points A & B. I would like to know why/how this was possible.
My thought process is of the following: We found the tangent at the general point therefore there must be some correlation as to how we are able to just sub the coordinates into the tangent.
The rectangular hyperbola H has equation xy=c2, where c is a constant. The point P(ct,tc is a general point on H.
a) Show that the tangent to H at P has equation
t2y+x=2ct
The tangents to H at the points A and B meet at the point (15c,−c)
b) Find, in terms of c, the coordinates of A and B
I'm only struggling with b)
The solution subs the x & y coordinate of the point (15c, -c) into t2y+x=2ct and cancels all the c's and has an equation for t and magically finds the points A & B. I would like to know why/how this was possible.
My thought process is of the following: We found the tangent at the general point therefore there must be some correlation as to how we are able to just sub the coordinates into the tangent.
The rectangular hyperbola H has equation xy=c2, where c is a constant. The point P(ct,tc) is a general point on H.
a) Show that the tangent to H at P has equation
t2y+x=2ct
The tangents to H at the points A and B meet at the point (15c,−c)
b) Find, in terms of c, the coordinates of A and B
I'm only struggling with b)
The solution subs the x & y coordinate of the point (15c, -c) into t2y+x=2ct and cancels all the c's and has an equation for t and magically finds the points A & B. I would like to know why/how this was possible.
My thought process is of the following: We found the tangent at the general point therefore there must be some correlation as to how we are able to just sub the coordinates into the tangent.
The rectangular hyperbola H has equation xy=c2, where c is a constant. The point P(ct,tc) is a general point on H.
a) Show that the tangent to H at P has equation
t2y+x=2ct
The tangents to H at the points A and B meet at the point (15c,−c)
b) Find, in terms of c, the coordinates of A and B
I'm only struggling with b)
The solution subs the x & y coordinate of the point (15c, -c) into t2y+x=2ct and cancels all the c's and has an equation for t and magically finds the points A & B. I would like to know why/how this was possible.
My thought process is of the following: We found the tangent at the general point therefore there must be some correlation as to how we are able to just sub the coordinates into the tangent.
The rectangular hyperbola H has equation xy=c2, where c is a constant. The point P(ct,tc) is a general point on H.
a) Show that the tangent to H at P has equation
t2y+x=2ct
The tangents to H at the points A and B meet at the point (15c,−c)
b) Find, in terms of c, the coordinates of A and B
I'm only struggling with b)
The solution subs the x & y coordinate of the point (15c, -c) into t2y+x=2ct and cancels all the c's and has an equation for t and magically finds the points A & B. I would like to know why/how this was possible.
My thought process is of the following: We found the tangent at the general point therefore there must be some correlation as to how we are able to just sub the coordinates into the tangent.
Since P represents any point on the curve, the tangent at P represents any tangent. If you enter certain values of x and y (into the equation for the "general tangent") you'll be able to find the corresponding values of c and t (which are used to express any point P on the curve). In this case the values of x and y (at the intersection point) correspond to the values of c and t that make up the coordinates of A and B.
Yes, I think. t is a parameter. solving the quadratic equation will find the values of t for A & B. But what I don't understand is why they subbed in the point (15c,-c) in the tangent? How was they able to do this?
Yes, I think. t is a parameter. solving the quadratic equation will find the values of t for A & B. But what I don't understand is why they subbed in the point (15c,-c) in the tangent? How was they able to do this?
The tangent equation is a function which outputs coordinates like (15c,-c). Recall that y=x^2, for example, will have coordinates like (0,0) and (2,4) lying on it. If you were told that y=ax^2 and were given the coordinate (2,8) then you'd be able to sub in and determine that a=2.
The rectangular hyperbola H has equation xy=c2, where c is a constant. The point P(ct,tc) is a general point on H.
a) Show that the tangent to H at P has equation
t2y+x=2ct
The tangents to H at the points A and B meet at the point (15c,−c)
b) Find, in terms of c, the coordinates of A and B
I'm only struggling with b)
The solution subs the x & y coordinate of the point (15c, -c) into t2y+x=2ct and cancels all the c's and has an equation for t and magically finds the points A & B. I would like to know why/how this was possible.
My thought process is of the following: We found the tangent at the general point therefore there must be some correlation as to how we are able to just sub the coordinates into the tangent.
No worries, thank you. I'll be using this little fact to destroy other questions, so it's all good. It's always good to find out why you're doing something, instead of just applying recipes to questions that look similar to one another
No worries, thank you. I'll be using this little fact to destroy other questions, so it's all good. It's always good to find out why you're doing something, instead of just applying recipes to questions that look similar to one another
Couldn't agree more , it's great that you want to understand the content rather than just learn it all off
Yes, I think. t is a parameter. solving the quadratic equation will find the values of t for A & B. But what I don't understand is why they subbed in the point (15c,-c) in the tangent? How was they able to do this?
Because it lies on both tangents for two some values of t.