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If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 from any point (2t^2, t^2), then. (A) t= -√2 1 (B) t=2 (C) t=2 √2 (D) none of these?
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If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 f...
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If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 f...
Given:
The equation of the hyperbola is x^2/2 - y^2/1 = 1.
To Find:
The value of t for which tangents of slope 1 can be drawn to the hyperbola from any point (2t^2, t^2).
Solution:
Step 1: Finding the Slope of Tangent
The slope of the tangent to the hyperbola at any point (x, y) is given by the derivative of the equation of the hyperbola with respect to x.
Differentiating the equation of the hyperbola x^2/2 - y^2/1 = 1 with respect to x, we get:
d(x^2/2)/dx - d(y^2/1)/dx = d(1)/dx
x/1 - 2y(1)/1 = 0
x - 2y = 0
x = 2y
The slope of the tangent is given by the derivative dy/dx.
Differentiating the equation x = 2y with respect to x, we get:
1 = 2(dy/dx)
(dy/dx) = 1/2
Step 2: Finding the Slope of Tangent from Point (2t^2, t^2)
Substituting x = 2t^2 and y = t^2 in the equation x = 2y, we have:
2t^2 = 2(t^2)
t^2 = t^2
The point (2t^2, t^2) lies on the line x = 2y.
The slope of the tangent from the point (2t^2, t^2) is given by the slope of the line x = 2y, which is 1/2.
Step 3: Comparing Slopes
We are given that the slope of the tangent is 1.
Comparing the slope of the tangent (1) with the slope of the tangent from the point (2t^2, t^2) (1/2), we can see that they are not equal.
Therefore, there is no value of t for which tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1 = 1 from any point (2t^2, t^2).
Answer:
(D) none of these
The equation of the hyperbola is x^2/2 - y^2/1 = 1.
To Find:
The value of t for which tangents of slope 1 can be drawn to the hyperbola from any point (2t^2, t^2).
Solution:
Step 1: Finding the Slope of Tangent
The slope of the tangent to the hyperbola at any point (x, y) is given by the derivative of the equation of the hyperbola with respect to x.
Differentiating the equation of the hyperbola x^2/2 - y^2/1 = 1 with respect to x, we get:
d(x^2/2)/dx - d(y^2/1)/dx = d(1)/dx
x/1 - 2y(1)/1 = 0
x - 2y = 0
x = 2y
The slope of the tangent is given by the derivative dy/dx.
Differentiating the equation x = 2y with respect to x, we get:
1 = 2(dy/dx)
(dy/dx) = 1/2
Step 2: Finding the Slope of Tangent from Point (2t^2, t^2)
Substituting x = 2t^2 and y = t^2 in the equation x = 2y, we have:
2t^2 = 2(t^2)
t^2 = t^2
The point (2t^2, t^2) lies on the line x = 2y.
The slope of the tangent from the point (2t^2, t^2) is given by the slope of the line x = 2y, which is 1/2.
Step 3: Comparing Slopes
We are given that the slope of the tangent is 1.
Comparing the slope of the tangent (1) with the slope of the tangent from the point (2t^2, t^2) (1/2), we can see that they are not equal.
Therefore, there is no value of t for which tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1 = 1 from any point (2t^2, t^2).
Answer:
(D) none of these
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If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 from any point (2t^2, t^2), then. (A) t= -√2 1 (B) t=2 (C) t=2 √2 (D) none of these?
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If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 from any point (2t^2, t^2), then. (A) t= -√2 1 (B) t=2 (C) t=2 √2 (D) none of these? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 from any point (2t^2, t^2), then. (A) t= -√2 1 (B) t=2 (C) t=2 √2 (D) none of these? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 from any point (2t^2, t^2), then. (A) t= -√2 1 (B) t=2 (C) t=2 √2 (D) none of these?.
If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 from any point (2t^2, t^2), then. (A) t= -√2 1 (B) t=2 (C) t=2 √2 (D) none of these? for Class 11 2024 is part of Class 11 preparation. The Question and answers have been prepared according to the Class 11 exam syllabus. Information about If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 from any point (2t^2, t^2), then. (A) t= -√2 1 (B) t=2 (C) t=2 √2 (D) none of these? covers all topics & solutions for Class 11 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If tangents of slope 1 can be drawn to the hyperbola x^2/2 - y^2/1=1 from any point (2t^2, t^2), then. (A) t= -√2 1 (B) t=2 (C) t=2 √2 (D) none of these?.
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