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The equation 2x2 – 3xy + 5y2 + 6x – 3y + 5 = 0 represents.
- a)A parabola
- b)An ellipse
- c)A hyperbola
- d)A pair of straight lines
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
The equation 2x2– 3xy + 5y2+ 6x – 3y + 5 = 0 represents.a)...
Comparing the equation with the standard form ax2+2hxy+by2+2gx+2fy+c=0
a=2,h=−3/2,b=5,g=3,f=−3/2,c=5
Δ=abc+2fgh−af2−bg2−ch2
=(2)(5)(5)+2(−3/2)(3)(−3/2)−(2)(−3/2)2−(5)(3)2−(5)(−3/2)2
=50+27/2−9/2−45−225/4
=−169/4 is not equal to 0
Descriminant =h2−ab
=(−3/2)2−(2)(5)
= 9/4−10
= −31/4<0
So, the curve represents either a circle or an ellipse
a is not equal to b and
Δ/a+b = −(169/4)/2+5
=−169/28<0
So, the curve represents a ellipse.
a=2,h=−3/2,b=5,g=3,f=−3/2,c=5
Δ=abc+2fgh−af2−bg2−ch2
=(2)(5)(5)+2(−3/2)(3)(−3/2)−(2)(−3/2)2−(5)(3)2−(5)(−3/2)2
=50+27/2−9/2−45−225/4
=−169/4 is not equal to 0
Descriminant =h2−ab
=(−3/2)2−(2)(5)
= 9/4−10
= −31/4<0
So, the curve represents either a circle or an ellipse
a is not equal to b and
Δ/a+b = −(169/4)/2+5
=−169/28<0
So, the curve represents a ellipse.
Most Upvoted Answer
The equation 2x2– 3xy + 5y2+ 6x – 3y + 5 = 0 represents.a)...
Given equation: 2x^2 + 3xy + 5y^2 + 6x + 3y + 5 = 0
To determine the shape represented by the given equation, we can analyze the coefficients and the degree of the equation.
1. Degree of the Equation
The degree of an equation is determined by the highest power of the variables present in the equation. In this case, the highest power of x and y is 2, which means the equation is of degree 2.
2. Coefficients of x^2 and y^2
The coefficients of x^2 and y^2 terms in the given equation are both positive. This indicates that the equation is not a hyperbola, as hyperbolas have opposite signs for these coefficients.
3. Coefficient of xy
The coefficient of xy term in the given equation is positive. This means that the equation is not an ellipse, as ellipses have a negative coefficient for the xy term.
4. Discriminant
To further analyze the equation, we can calculate the discriminant. The discriminant can help us determine the nature of the roots of the equation and provide information about its conic section.
The equation of a conic section can be written as:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
The discriminant (D) can be calculated using the formula:
D = B^2 - 4AC
In this case, A = 2, B = 3, and C = 5. Calculating the discriminant:
D = (3)^2 - 4(2)(5) = 9 - 40 = -31
5. Nature of the Roots
The nature of the roots of the given equation can be determined based on the value of the discriminant (D).
If D > 0, the equation represents a pair of intersecting lines.
If D = 0, the equation represents a pair of coincident lines.
If D < 0,="" the="" equation="" represents="" a="" conic="" />
In this case, D = -31, which is less than 0. Therefore, the given equation represents a conic section.
6. Shape of the Conic Section
Since the coefficients of x^2 and y^2 terms are both positive, and the coefficient of the xy term is positive, the given equation represents an ellipse. The positive coefficients indicate an elongated shape, and the positive coefficient of xy indicates that the ellipse is tilted or rotated.
Therefore, the correct answer is option 'B' - An ellipse.
To determine the shape represented by the given equation, we can analyze the coefficients and the degree of the equation.
1. Degree of the Equation
The degree of an equation is determined by the highest power of the variables present in the equation. In this case, the highest power of x and y is 2, which means the equation is of degree 2.
2. Coefficients of x^2 and y^2
The coefficients of x^2 and y^2 terms in the given equation are both positive. This indicates that the equation is not a hyperbola, as hyperbolas have opposite signs for these coefficients.
3. Coefficient of xy
The coefficient of xy term in the given equation is positive. This means that the equation is not an ellipse, as ellipses have a negative coefficient for the xy term.
4. Discriminant
To further analyze the equation, we can calculate the discriminant. The discriminant can help us determine the nature of the roots of the equation and provide information about its conic section.
The equation of a conic section can be written as:
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
The discriminant (D) can be calculated using the formula:
D = B^2 - 4AC
In this case, A = 2, B = 3, and C = 5. Calculating the discriminant:
D = (3)^2 - 4(2)(5) = 9 - 40 = -31
5. Nature of the Roots
The nature of the roots of the given equation can be determined based on the value of the discriminant (D).
If D > 0, the equation represents a pair of intersecting lines.
If D = 0, the equation represents a pair of coincident lines.
If D < 0,="" the="" equation="" represents="" a="" conic="" />
In this case, D = -31, which is less than 0. Therefore, the given equation represents a conic section.
6. Shape of the Conic Section
Since the coefficients of x^2 and y^2 terms are both positive, and the coefficient of the xy term is positive, the given equation represents an ellipse. The positive coefficients indicate an elongated shape, and the positive coefficient of xy indicates that the ellipse is tilted or rotated.
Therefore, the correct answer is option 'B' - An ellipse.
Community Answer
The equation 2x2– 3xy + 5y2+ 6x – 3y + 5 = 0 represents.a)...
B is correct answer
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The equation 2x2– 3xy + 5y2+ 6x – 3y + 5 = 0 represents.a)A parabolab)An ellipsec)A hyperbolad)A pair of straight linesCorrect answer is option 'B'. Can you explain this answer?
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