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Q01 |
- Diagram of 2*x^2 + 4*y^2 + 4*x - 4*y - 9 = 0
- Q02 | - Locus of 2*x^2 + 4*y^2 + 4*x - 8*y - 10 = 0
- Q03 | - Locus of 2*x^2 + 4*y^2 + 4*x - 8*y + 6 = 0
- Q04 | - Locus of 2*x^2 + 4*y^2 + 4*x - 8*y + 12 = 0
- Q05 | - Locus of A*x^2 + C*y^2 + D*x + E*y + F = 0
- Q06 | - Equations of ellipse
- Q07 | - Sketch ellipse of F(x, y)
- Q02 | - Locus of 2*x^2 + 4*y^2 + 4*x - 8*y - 10 = 0
Q01. Diagram : 2*x^2 + 4*y^2 + 4*x - 4*y - 9 = 0
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Q02. Locus of 2*x^2 + 4*y^2 + 4*x - 8*y - 10 = 0
Change to standard form using completing the square
- 2*(x^2 + 2*x + 1 - 1) + 4*(y^2 - 2*y + 1 - 1) - 10 = 0
- 2*(x + 1)^2 + 4*(y - 1)^2 - 2 - 4 - 10 = 0
- 2*(x + 1)^2 + 4*(y - 1)^2 = 16
- ((x + 1)/a)^2 + ((y - 1)/b)^2 = 1
- Hence a = Sqr(8) and b = Sqr(4)
- Focal length = f = Sqr(a^2 - b^2) = Sqr(8 - 4) = 2
- It is an ellipse
- The center is at (-1, 1)
- Semi-axese a = Sqr(8) and b = Sqr(4)
- Focal length = f = Sqr(a^2 - b^2) = Sqr(4) = 2
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Q03. Locus of 2*x^2 + 4*y^2 + 4*x - 8*y + 8 = 0
Change to standard form using completing the square
- 2*(x^2 + 2*x + 1 - 1) + 4*(y^2 - 2*y + 1 - 1) + 6 = 0
- 2*(x + 1)^2 + 4*(y - 1)^2 - 2 - 4 + 6 = 0
- 2*(x + 1)^2 + 4*(y - 1)^2 = 0
- It is a point at (-1, 1)
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Q04. Locus of 2*x^2 + 4*y^2 + 4*x - 8*y + 12 = 0
Change to standard form using completing the square
- 2*(x^2 + 2*x + 1 - 1) + 4*(y^2 - 2*y + 1 - 1) + 12 = 0
- 2*(x + 1)^2 + 4*(y - 1)^2 - 2 - 4 + 12 = 0
- 2*(x + 1)^2 + 4*(y - 1)^2 = -6
- The equation is illegal in the real number system
- Hence 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 ellipse
1. Standard form in rectangular form
- Equation : ((x - h)/a)^2 + ((y - k)/b)^2 = 1
- Center at (h, k)
- Principal axis is x = h if a > b
- Principal axis is y = k if b > a
- Focal length = f = Sqr(a^2 - b^2)
- Directrix : D*e = (b^2)/a
- Equation : F(x, y) = A*x^2 + C*y^2 + D*x + E*y + F = 0
- A and C must be same signs
- Change to standard form for interpretation
- Equation : x = h + a*cos(A) and y = k + b*sin(A)
- Use Pythagorean relation to change it into standard form for interpretation
- Center at (h, k)
- Sem-axese are a and b
- Focal length = f = Sqr(a^2 - b^2)
- R = (D*e)/(1 - e*sin(A))
- R = (D*e)/(1 + e*sin(A))
- R = (D*e)/(1 - e*cos(A))
- R = (D*e)/(1 + e*cos(A))
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Q07. Sketch ellipse of F(x, y)
Sketch diagram of 2*x^2 + 4*y^2 + 4*x - 4*y - 9 = 0
- Star the QB version program
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Program |
ZM34 : Option 10
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Program |
ZM34 : Option 10
- Give input on keyboard
- Type option 10 and press return
- Type 2 and press return
- Type 2, 4, 4, -4, -9 and press return
- Use the diagram
- 1. Find center
- 2. Find semi-axese
- 3. Find focal length
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