Hi again 
I'm still in an exam period, next up being my 4-unit maths half-yearly. Thankyou to everyone who posted in my relativity thread, all information was greatly appreciated. From what I can gather, I think I did pretty well in that exam
Anyway, I've been doing hyperbola and ellipse questions all weekend, and have had difficulty for any problems involving points of predetermined relationships. For example:
A variable chord PQ of the ellipse x^2/a^2 + y^2/b^2 = 1 subtends a right angle at the point (a, 0). Show that PQ passes through a fixed point of the X-axis.
I got so far as to extablish the perpendicular relationship between the P(x1, y1)R(a, 0) and Q(x2, y2)R(a, 0), but couldn't derive the equation necessary for elimination of the second set of coordinates. I have had similar problems with questions regarding focal chords. I have no trouble in determining the locus of intersections between tangents, finding the areas of triangles produced by tangents to a locus, etc. Here is my working thus far:
P(x1, y1) Q(x2, y2) R(a, 0)
mQR = y2 / (x2 - a)
mPR = y1 / (x1 - a)
mPR.mQR = -1
(y1 . y2) / ((x1 - a)(x2 - a)) = - 1
y1y2 = - x1x2 + a(x1 + x2) - a^2
mPQ = (y2 - y1)/(x2 - x1)
y - y1 = (y2 - y1)/(x2 - x1) * (x - x1)
PQ intersects the x axis at T
y = 0
which solves for
x = (x1y2 - y1x2)/(y2 - y1)
using the result from perpendicular lines, x1 = (ax2 - a^2 - y1y2)/(x2 - a)
substituting this into x for T
x = (ax2y2 - y2a^2 - y1(y2)^2 -y1(x2)^2 + ay1x2)/((x2 - a)(y2 - y1))
Which I'm sure you can see, is an ockham's razor solution.
Sorry for all the questions. Any response is much appreciated
I'm still in an exam period, next up being my 4-unit maths half-yearly. Thankyou to everyone who posted in my relativity thread, all information was greatly appreciated. From what I can gather, I think I did pretty well in that exam
Anyway, I've been doing hyperbola and ellipse questions all weekend, and have had difficulty for any problems involving points of predetermined relationships. For example:
A variable chord PQ of the ellipse x^2/a^2 + y^2/b^2 = 1 subtends a right angle at the point (a, 0). Show that PQ passes through a fixed point of the X-axis.
I got so far as to extablish the perpendicular relationship between the P(x1, y1)R(a, 0) and Q(x2, y2)R(a, 0), but couldn't derive the equation necessary for elimination of the second set of coordinates. I have had similar problems with questions regarding focal chords. I have no trouble in determining the locus of intersections between tangents, finding the areas of triangles produced by tangents to a locus, etc. Here is my working thus far:
P(x1, y1) Q(x2, y2) R(a, 0)
mQR = y2 / (x2 - a)
mPR = y1 / (x1 - a)
mPR.mQR = -1
(y1 . y2) / ((x1 - a)(x2 - a)) = - 1
y1y2 = - x1x2 + a(x1 + x2) - a^2
mPQ = (y2 - y1)/(x2 - x1)
y - y1 = (y2 - y1)/(x2 - x1) * (x - x1)
PQ intersects the x axis at T
y = 0
which solves for
x = (x1y2 - y1x2)/(y2 - y1)
using the result from perpendicular lines, x1 = (ax2 - a^2 - y1y2)/(x2 - a)
substituting this into x for T
x = (ax2y2 - y2a^2 - y1(y2)^2 -y1(x2)^2 + ay1x2)/((x2 - a)(y2 - y1))
Which I'm sure you can see, is an ockham's razor solution.
Sorry for all the questions. Any response is much appreciated
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