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Q01 |
- Diagram of x^2 + y^2 + 4*x - 4*y - 9 = 0
- Q02 | - Locus of x^2 + y^2 + 4*x - 4*y - 17 = 0
- Q03 | - Locus of x^2 + y^2 + 4*x - 4*y + 8 = 0
- Q04 | - Locus of x^2 + y^2 + 4*x - 4*y + 10 = 0
- Q05 | - Locus of A*x^2 + C*y^2 + D*x + E*y + F = 0
- Q06 | - Equations of circle
- Q07 | - Sketch circle of F(x, y) = 0
- Q02 | - Locus of x^2 + y^2 + 4*x - 4*y - 17 = 0
Q01. Diagram : x^2 + y^2 + 4*x - 4*y - 9 = 0
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Q02. Locus of x^2 + y^2 + 4*x - 4*y - 17 = 0
Change to standard form using completing the square
- (x^2 + 4*x + 4 - 4) + (y^2 - 4*y + 4 - 4) - 17 = 0
- (x + 2)^2 + (y - 2)^2 - 4 - 4 - 17 = 0
- (x + 2)^2 + (y - 2)^2 = (5)^2
- It is a circle
- The center is at (-2, 2)
- Radius is equal 5
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Q03. Locus of x^2 + y^2 + 4*x - 4*y + 8 = 0
Change to standard form using completing the square
- (x^2 + 4*x + 4 - 4) + (y^2 - 4*y + 4 - 4) + 8 = 0
- (x + 2)^2 + (y - 2)^2 - 4 - 4 + 8 = 0
- (x + 2)^2 + (y - 2)^2 = 0
- It is a point at (-2, 2)
- Radius is equal 0
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Q04. Locus of x^2 + y^2 + 4*x - 4*y + 10 = 0
Change to standard form using completing the square
- (x^2 + 4*x + 4 - 4) + (y^2 - 4*y + 4 - 4) + 10 = 0
- (x + 2)^2 + (y - 2)^2 - 4 - 4 + 10 = 0
- (x + 2)^2 + (y - 2)^2 = -2
- It is not existed in real number system
- There is no locus
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Q05. Locus of A*x^2 + C*y^2 + D*x + E*y + F = 0
The locus
- It is a circle if A = C
- It is an ellipse if A and C have same sign
- It is a hyprbola if A and C have different sign
- Standard form A*(x - h)^2 + C*(y - k)^2 = p
- The locus when A = C
- it is a circle if p > 0
- it is a point if p = 0
- it is not exited in real number system if p < 0
- The locus when A and C have same signs
- it is an ellipse if p > 0
- it is a point if p = 0
- it is not exited in real number system if p < 0
- The locus when A and C have different signs
- it is a hyperbola with principal axis y = k if p > 0
- it gives two lines if p = 0
- it is a hyperbola with different principal axis x = h if p < 0
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Q06. Equations of Circle
1. Standard form in rectangular coordinates
- Equation : (x - h)^2 + (y - k)^2 = r^2
- Center at (h, k)
- Radius = r
- Equation F(x, y) = x^2 + y^2 + D*x + E*y + F = 0
- We have to change it into standard form for interpretation
- Equation : x = h + r*cos(A) and y = k + r*sin(A)
- Use Pythagorean realtion we can change this equation to standard form
- Center at (h, k)
- Radius is r
- R = sin(A)
- Since R = Sqr(x^2 + y^2) and y = R*sin(A)
- Hence R = y/R
- Hence R^2 = y
- Hence x^2 + y^2 = y
- Hence x^2 + (y - 1/2)^2 = (1/2)^2
- Hence center at (0, 1/2)
- Hence radius = 1/2
- R = cos(A)
- Since R = Sqr(x^2 + y^2) and x = R*cos(A)
- Hence R = x/R
- Hence R^2 = x
- Hence x^2 + y^2 = x
- Hence (x - 1/2)^2 + y^2 = (1/2)^2
- Hence center at (1/2, 0)
- Hence radius = 1/2
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Q07. Sketch circle of F(x, y) = 0
Sketch diagram of x^2 + y^2 + 4*x - 4*y - 9 = 0
- Star the QB version program
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Program |
ZM34 : Option 9
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Program |
ZM34 : Option 9
- Give input on keyboard
- Type option 9 and press return
- Type 2 and press return
- Type 1, 1, 4, -4, -9 and press return
- Use the diagram
- 1. Find center
- 2. Find redius
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