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The equation 2x2 + 5xy – 12y2 = 0 represents a
- a)circle
- b)pair of non-perpendicular intersecting straight lines
- c)pair of perpendicular straight lines
- d)hyperbola
Correct answer is option 'B'. Can you explain this answer?
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The equation 2x2 + 5xy – 12y2 = 0 represents aa)circleb)pair of ...
Understanding the Equation
The given equation is 2x² + 5xy - 12y² = 0. This is a second-degree polynomial equation in two variables (x and y). To determine its geometric representation, we need to analyze the coefficients.
Identifying the General Form
The general form of a conic section is given by Ax² + Bxy + Cy² + Dx + Ey + F = 0.
In our case:
- A = 2
- B = 5
- C = -12
Discriminant Analysis
To classify the conic section, we calculate the discriminant (D) using the formula D = B² - 4AC.
- D = (5)² - 4(2)(-12)
- D = 25 + 96
- D = 121
Since D > 0, this indicates that the equation represents a pair of straight lines.
Classification of the Lines
Next, we determine whether these lines are intersecting and if they are perpendicular.
- For a pair of intersecting lines, we check the relation between A, B, and C using the following:
- If A * C < 0,="" the="" lines="" are="" />
- In our case:
- 2 * (-12) = -24, which is less than 0.
This confirms that the lines intersect.
Perpendicularity Check
To check if the lines are perpendicular, we consider the condition:
- If B² = 4AC, the lines are perpendicular.
Since B² = 25 and 4AC = 4 * 2 * (-12) = -96, we see that B² ≠ 4AC, confirming that the lines are not perpendicular.
Conclusion
Thus, the equation 2x² + 5xy - 12y² = 0 represents a pair of non-perpendicular intersecting straight lines, which corresponds to option 'B'.
The given equation is 2x² + 5xy - 12y² = 0. This is a second-degree polynomial equation in two variables (x and y). To determine its geometric representation, we need to analyze the coefficients.
Identifying the General Form
The general form of a conic section is given by Ax² + Bxy + Cy² + Dx + Ey + F = 0.
In our case:
- A = 2
- B = 5
- C = -12
Discriminant Analysis
To classify the conic section, we calculate the discriminant (D) using the formula D = B² - 4AC.
- D = (5)² - 4(2)(-12)
- D = 25 + 96
- D = 121
Since D > 0, this indicates that the equation represents a pair of straight lines.
Classification of the Lines
Next, we determine whether these lines are intersecting and if they are perpendicular.
- For a pair of intersecting lines, we check the relation between A, B, and C using the following:
- If A * C < 0,="" the="" lines="" are="" />
- In our case:
- 2 * (-12) = -24, which is less than 0.
This confirms that the lines intersect.
Perpendicularity Check
To check if the lines are perpendicular, we consider the condition:
- If B² = 4AC, the lines are perpendicular.
Since B² = 25 and 4AC = 4 * 2 * (-12) = -96, we see that B² ≠ 4AC, confirming that the lines are not perpendicular.
Conclusion
Thus, the equation 2x² + 5xy - 12y² = 0 represents a pair of non-perpendicular intersecting straight lines, which corresponds to option 'B'.
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The equation 2x2 + 5xy – 12y2 = 0 represents aa)circleb)pair of non-perpendicular intersecting straight linesc)pair of perpendicular straight linesd)hyperbolaCorrect answer is option 'B'. Can you explain this answer?
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