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The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4=0. The third side which is variable always passes through the point (-5, -1) ,if the range of values of the slope of the third side so that the origin is an Interior point of the triangle is (a, b) then find a 1/b²?
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The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4...
Given information:
- The combined equation of the two sides of a triangle is x²–3y²–2xy 8y–4=0.
- The third side, which is variable, always passes through the point (-5, -1).
- We need to find the range of values for the slope of the third side so that the origin is an interior point of the triangle.
Step 1: Find the equation of the third side:
To find the equation of the third side, we need to find the equation of the line passing through the point (-5, -1) and the origin (0, 0). Let's denote the slope of this line as "m".
Using the slope-intercept form of a line, the equation of the line passing through (-5, -1) and (0, 0) can be written as:
y - (-1) = m(x - (-5))
y + 1 = m(x + 5)
y = mx + 5m - 1
Step 2: Substitute the equation of the third side into the combined equation:
Substituting the equation of the third side (y = mx + 5m - 1) into the combined equation of the two sides (x²–3y²–2xy 8y–4=0), we get:
x² – 3(mx + 5m - 1)² – 2x(mx + 5m - 1) + 8(mx + 5m - 1) – 4 = 0
Simplifying the above equation, we obtain:
(1 – 3m²)x² + (-6m + 8 – 30m + 40m² + 24m²)x + (-3 + 10m – 5m²) = 0
Step 3: Analyzing the equation:
The equation obtained in Step 2 represents a quadratic equation in terms of x. For the origin to be an interior point of the triangle, this quadratic equation should have two real and distinct solutions for x.
For a quadratic equation to have two real and distinct solutions, the discriminant (b² - 4ac) should be positive. In our equation, the coefficients of x², x, and the constant term are:
a = (1 – 3m²)
b = (-6m + 8 – 30m + 40m² + 24m²)
c = (-3 + 10m – 5m²)
The discriminant can be calculated as:
(b² - 4ac) > 0
Step 4: Calculate the discriminant:
Substituting the values of a, b, and c into the discriminant formula, we have:
((-6m + 8 – 30m + 40m² + 24m²)²) - 4(1 – 3m²)(-3 + 10m – 5m²) > 0
Simplifying the above inequality and solving for m, we get a range of values for m.
Step 5: Find the range of values for the slope:
Solving the inequality obtained in Step 4, we can find the range of values for m. Once
- The combined equation of the two sides of a triangle is x²–3y²–2xy 8y–4=0.
- The third side, which is variable, always passes through the point (-5, -1).
- We need to find the range of values for the slope of the third side so that the origin is an interior point of the triangle.
Step 1: Find the equation of the third side:
To find the equation of the third side, we need to find the equation of the line passing through the point (-5, -1) and the origin (0, 0). Let's denote the slope of this line as "m".
Using the slope-intercept form of a line, the equation of the line passing through (-5, -1) and (0, 0) can be written as:
y - (-1) = m(x - (-5))
y + 1 = m(x + 5)
y = mx + 5m - 1
Step 2: Substitute the equation of the third side into the combined equation:
Substituting the equation of the third side (y = mx + 5m - 1) into the combined equation of the two sides (x²–3y²–2xy 8y–4=0), we get:
x² – 3(mx + 5m - 1)² – 2x(mx + 5m - 1) + 8(mx + 5m - 1) – 4 = 0
Simplifying the above equation, we obtain:
(1 – 3m²)x² + (-6m + 8 – 30m + 40m² + 24m²)x + (-3 + 10m – 5m²) = 0
Step 3: Analyzing the equation:
The equation obtained in Step 2 represents a quadratic equation in terms of x. For the origin to be an interior point of the triangle, this quadratic equation should have two real and distinct solutions for x.
For a quadratic equation to have two real and distinct solutions, the discriminant (b² - 4ac) should be positive. In our equation, the coefficients of x², x, and the constant term are:
a = (1 – 3m²)
b = (-6m + 8 – 30m + 40m² + 24m²)
c = (-3 + 10m – 5m²)
The discriminant can be calculated as:
(b² - 4ac) > 0
Step 4: Calculate the discriminant:
Substituting the values of a, b, and c into the discriminant formula, we have:
((-6m + 8 – 30m + 40m² + 24m²)²) - 4(1 – 3m²)(-3 + 10m – 5m²) > 0
Simplifying the above inequality and solving for m, we get a range of values for m.
Step 5: Find the range of values for the slope:
Solving the inequality obtained in Step 4, we can find the range of values for m. Once
Community Answer
The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4...
Is it 1/5?
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The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4=0. The third side which is variable always passes through the point (-5, -1) ,if the range of values of the slope of the third side so that the origin is an Interior point of the triangle is (a, b) then find a 1/b²?
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The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4=0. The third side which is variable always passes through the point (-5, -1) ,if the range of values of the slope of the third side so that the origin is an Interior point of the triangle is (a, b) then find a 1/b²? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4=0. The third side which is variable always passes through the point (-5, -1) ,if the range of values of the slope of the third side so that the origin is an Interior point of the triangle is (a, b) then find a 1/b²? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4=0. The third side which is variable always passes through the point (-5, -1) ,if the range of values of the slope of the third side so that the origin is an Interior point of the triangle is (a, b) then find a 1/b²?.
The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4=0. The third side which is variable always passes through the point (-5, -1) ,if the range of values of the slope of the third side so that the origin is an Interior point of the triangle is (a, b) then find a 1/b²? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4=0. The third side which is variable always passes through the point (-5, -1) ,if the range of values of the slope of the third side so that the origin is an Interior point of the triangle is (a, b) then find a 1/b²? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The two sides of a triangle have the combined equation x²–3y²–2xy 8y–4=0. The third side which is variable always passes through the point (-5, -1) ,if the range of values of the slope of the third side so that the origin is an Interior point of the triangle is (a, b) then find a 1/b²?.
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