-
Q01 |
- Diagram : R = (D*e)/(1 - e*cos(A))
- Q02 | - Directrix
- Q03 | - Convert (x/a)^2 - (y/b)^2 = 1 to polar form
- Q04 | - Convert R = (D*e)/(1 - e*cos(A)) to rectangular form
- Q05 | - Convert (x/4)^2 - (y/3)^2 = 1 to polar form
- Q06 | - Convert R = 2.25/(1 + 1.25*cos(A)) to rectangular form
- Q07 | - Reference
- Q02 | - Directrix
Q01. Diagram :
|
Go to Begin
Q02. Directrix
Defintion
- R = point P(x, y) to focus (0, 0)
- D = distance from focus to directrix
- D + x = distance from P to directrix
- R/(R + x) = e is hperbola if e > 1
- Where e = f/a
- Since x = R*cos(A)
- Hence R = (D*e)/(1 - e*cos(A))
- From diagram we see that FU = f - a
- When A = 180, R = FU = (D*e)/(1 + e)
- Hence D*e = (f - a)*(1 + e)
- Since e = f/a
- Hence D*e = (f - a)*(1 + f/a)
- Hence D*e = (f^2 - a^2)/a
- Hence D*e = (b^2)/a
Go to Begin
Q03. Convert (x/a)^2 - (y/b)^2 = 1 to polar form
Method 1 : Find D and e
- Corresponding form
- R = (D*e)/(1 - e*cos(A)) : The center at (f, 0)
- ((x - f)/a)^2 + (y/b)^2 = 1 : The center at (f, 0)
- (x/a)^2 + (y/b)^2 = 1 : The center at (0, 0)
- Since the translation from (0, 0) to (f, 0), the diagram is no change
- Hence we know that the polar form is R = (D*e)/(1 - e*cos(A))
- R = D*e/(1 - e*cos(A))
- The focal point is at F(0, 0)
- The center is at (f, 0)
- Hence rectangular form is ((x - f)/a)^2 - (y/b)^2 = 1
- Remove denominator and simplify
- (b^2)*(x + f)^2 + (a^2)*(y^2) = (a^2)*b*2.
- (b^2)*(x^2) + (b^2)*x*f + (b^2)*(f^2) + (a^2)*(y^2) = (a^2)*(b^2)
- Since b^2 = f^2 - a^2.
- Hence (b^2)*(x^2) + (a^2)*(y^2) + 2*(b^2)*x*f = (a^2)*(b^2) - (b^2)*(f^2)
- -(a^2 - f^2)*(x^2) + (a^2)*(y^2) + 2*(b^2)*x*f = -(b^2)*(a^2 - f^2)
- Polar coordiantes
- x = R*cos(A)
- y = R*sin(A)
- R^2 = x^2 + y^2
- Substitute polar coordinates into above equation
- (x^2 + y^2)*(a^2) - (f^2)*(x^2) + 2*(b^2)*x*f = b^4
- (R^2)*(a^2) - (f^2)*(x^2) + 2*(b^2)*x*f - b^4 = 0
- (R^2)*(a^2) - ((f^2)*(x^2) - 2*(b^2)*x*f + b^4) = 0
- (R^2)*(a^2) - (f*x - b^2)^2 = 0
- R*a = f*x - b^2
- Since x = R*cos(A)
- Hence R*a - R*f*cos(A) = b^2
- R*(a - f*cos(A)) = b^2
- R*(1 - f*cos(A)/a) = (b^2)/a
- Since e = f/a and D*e = (b^2)/a
- Hence R = (D*e)/(1 - e*cos(A))
Go to Begin
Q04. Convert R = (D*e)/(1 - e*cos(A)) to rectangular form
Solution : Use D
- The corresponding form
- Rectangular form : ((x - f)/a)^2 - (y/b)^2 = 1
- Polar form : R = (D*e)/(1 - e*cos(A))
- Since center at (0, 0) : (x/a)^2 - (y/b)^2 = 1 ........... (1)
- Center at (f, 0) : ((x - f)/a)^2 - (y/b)^2 = 1 ........... (2)
- Polar form center at (f, 0) : R = (D*e)/(1 - e*cos(A)) .. (3)
- Since (1) and (2) give congruent diagram
- Hence we can translate center from (1) to (2)
- From (2) and (3), we see that (2) is the answer
- Also (1) is the answer
- R = D*e/(1 - e*cos(A))
- R*(1 - e*cos(A)) = D*e
- R - e*R*cos(A) = D*e
- R = D*e + e*x
- Since D*e = (b^2)/a and e = f/a
- Hence R = (b^2)/a + f*x/a
- Or a*R = b^2 + x*f
- Square both sides
- (a^2)*(R^2) = (b^2 + x*f)^2
- (a^2)*(R^2) = b^4 + 2*(b^2)*x*f + (x^2)*(f^2)
- (a^2)*(x^2 + y^2) - 2*(b^2)*x*f - (x^2)*(f^2) = b^4
- (a^2)*(x^2) + (a^2)*(y^2) - 2*(b^2)*x*f - (x^2)*(f^2) = b^4 .......... (1)
- Since b^2 = f^2 - a^2, (1) becomes
- (a^2 - f^2)*(x^2) + (a^2)*(y^2) - 2*(b^2)*x*f = (b^2)*(a^2 - f^2)
- -(b^2)*(x^2 - 2*x*f + f^2) + (a^2)*(y^2) = -(b^2)*(a^2) ............. (2)
- (b^2)*(x - f)^2 - (a^2)*(y^2) = (a^2)*(b^2)
- Divide both sides by (a^2)*(b^2)
- ((x - f)/a)^2 - (y/b)^2 = 1
Go to Begin
Q05. Convert(x/4)^2 - (y/3)^2 = 1 to polar form
Method 1 : Find D and e
- The polar form is R = (D*e)/(1 - e*cos(A))
- Since f = Sqr(a^2 + b^2) = 5
- Also D*e = (f - a)*(1 + e) = (f^2 - a^2)/a = (b^2)/a
- a = 4 and b = 3
- e = f/a = 5/4 = 1.25
- D*e = (5 - 4)*(1 + 1.25) = 2.25
- R = 2.25/(1 - 1.25*cos(A))
- Since center of R = (D*e)/(1 - e*cos(A)) is at (f, 0)
- We must use equation ((x - f)/a)^2 - (y/b)^2 = 1
- Since f = Sqr(a^2 + b^2) = 5
- Hence 9*(x - 5)^2 - 16*(y^2) = 144
- 9*(x^2) - 90*x + 9*25 - 16*(y^2) = 144
- Change coefficient of x^2 same as y^2
- -16*x^2 + 25*x^2 - 90*x - 16*(y^2) = 144 - 225
- -16*(x^2 + y^2) = -25*(x^2) + 90*x - 81
- Since R^2 = x^2 + y^2
- Hence 16*(R^2) = (5*x - 9)^2
- Take square root on both sides : 4*R = 5*x - 9
- R = 1.25*x + 2.25
- Since x = R*cos(A)
- Hence R = 1.25*R*cos(A) - 2.25
- The answer is R = 2.25/(1 + 1.25*cos(A))
Go to Begin
Q06. Convert R = 2.25/(1 + 1.25*cos(A)) to rectangular form
Solution : Use D*e = (f^2 - a^2)/a = (b^2)/a and e = f/a
- The corresponding form
- ((x - f)/a)^2 + (y/b)^2 = 1
- (x/a)^2 + (y/b)^2 = 1
- R = (D*e)/(1 + e*cos(A))
- For R = 2.25/(1 + 1.25*cos(A))
- e = 1.25
- D*e = 2.25
- e = f/a and f = 1.25*a ...... (1)
- D*e = (f^2 - a^2)/a
- 2.25*a = (f^2 - a^2) ........ (2)
- Substitute (1) into (2), we have
- 2.25*a = (1.25*a)^2 - a^2)
- 2.25*a = 0.5625*a^2
- a*(0.5625*a - 2.25) = 0
- Hence a = 0 or a = 2.25/0.5625 = 4
- Substitute a = 4 into (1), Hence f = 5
- Since f^2 = a^2 + b^2
- Hence b^2 = f^2 - a^2 = 25 - 16 = 9
- Hence b = 3
- Hence equarion is (x/5)^2 - (y/3)^2 = 1
- General solution : ((x - h)/a)^2 - ((y - k)/b)^2 = 1
- R = 2.25/(1 + 1.25*cos(A))
- R*(1 + 1.25*cos(A)) = 2.25
- Since R^2 = x^2 + y^2 and x = R*cos(A)
- Hence R + 1.25*x = 2.25 or R = 2.25 - 1.25*x ......... (1)
- Square both sides of (1)
- x^2 + y^2 = 2.25^2 + 2*2.25*1.25*x + 1.5625*x^2
- Simply 0.5625*x^2 + 5.625*x - y^2 + 5.0625 = 0 ....... (2)
- Use completing the square, (2) becomes
- 0.5625*(x^2 + 10*x + 25 - 25) - y^2 = -5.0625
- 0.5625*(x - 5)^2 - y^2 = -5.0625 + 25*0.5625
- 0.5625*(x - 5)^2 - y^2 = 9
- ((x - 4)/4)^2 - (y/3)^2 = 1
Go to Begin
Q07. Reference
Reference
-
Subject |
Hyperbola
Go to Begin