Figure 119 : Three points A(7,4), B(3,1), C(0,k)
Q01. Diagram
|
Coordinate geometry 
|
Go to Begin
Q02. A(7,4), B(3,1), C(0,k) if AC + BC is minimum find k
Keywords
- y is minimu if y' = 0
- if y = a*x^2 + b*x + c, y' = 2*a*x + b
- If y = Sqr(a*x^2 + b*x + c), y' = (2*a*x + b)/(2*Sqr(a*x^2 + b*x + c)
- 1. Draw rectangular coordinate
- 2. Draw points A, B and C
- Let h = AC + BC
- = Sqr((7 - 0)^2 + (4 - k)^2) + Sqr((3 - 0)^2 + (1 - k)^2)
- = Sqr(49 + 16 - 8*k + k^2) + Sqr(9 + 1 - 2*k + k^2)
- = Sqr(k^2 - 8*k + 65) + Sqr(k^2 - 2*k + 10)
- If h is minimum
- dh/dk = 0
- (2*k - 8)/(2*Sqr(k^2 - 8*k + 65)) + (2*k - 2)/(2*Sqr(k^2 - 2*k + 10)) = 0
- Simplify, we have
- (k - 4)*Sqr(k^2 - 2*k + 10) = -(k - 1)*Sqr(k^2 - 8*k + 65)
- Square both sides
- ((k - 4)^2)*(9 + (k - 1)^2) = ((k - 1)^2)*(49 + (k^2 - 4)^2)
- 9*(k - 4)^2 + ((k - 4)*(x - 1))^2 = 49*((k - 1)^2) + ((k - 1)*(x^2 - 4))^2
- 9*(k - 4)^2 = 49*(k - 1)^2
- Take square on both sides
- 3*(k - 4) = +7*(k - 1), 4*k = -5 or k = -1.25
- 3*(k - 4) = -7*(k - 1), 10*k = 19 or k = 1.9
Go to Begin
Q03. A(7,4), B(3,1), C(0,k) If AC^2 + BC^2 is minimum, find k
Keywords
- Distance formula
- Completing the square
- h^2 = AC^2 + BC^2
- h^2 = (49 + (4 - k)^2) + (9 + (k - 1)^2)
- h^2 = 65 - 8*k + k^2 + 10 + k^2 - 2*k
- h^2 = 2*k^2 - 10*k + 75
- Use completing the square
- h^2 = 2*(k^2 - 5*k + (5/2)^2 - (5/2)^2) + 75
- h^2 = 2*(k - 5/2)^2 + 75 - 2*(5/2)^2
- h^2 is minimum if k = 5/2
- Use derivetative
- d(h^2)/dk = 0
- 4*k - 10 = 0
- k = 2.5
Go to Begin
Q04. Study questions
- 1. If y = a*x^2 + b*x + c, how to find the minimum ?
- 2. Give the distance formula between two points (x1,y1) and (x2,y2)
- 3. What is the completing square in quadratic function ?
Go to Begin