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The equation of a tangent to the parabola y2 = 8x is y = x + 2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is [2007]
- a)(2, 4)
- b)(–2, 0)
- c)(–1, 1)
- d)(0, 2)
Correct answer is option 'B'. Can you explain this answer?
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To find the point on the line y = x^2 from which the other tangent to the parabola y^2 = 8x is perpendicular to the given tangent, we need to find the point of tangency for the given tangent.
The given tangent is y = x^2. To find the slope of this tangent, we take the derivative of y with respect to x:
dy/dx = 2x
This gives us the slope of the tangent at any point (x, x^2) on the parabola.
The slope of a perpendicular line is the negative reciprocal of the given slope. So, the slope of the perpendicular tangent is -1/(2x).
For the perpendicular tangent to be perpendicular to the given tangent, the product of their slopes should be -1:
(2x)*(-1/(2x)) = -1
Simplifying, we get:
-1 = -1
This is true for any value of x, so the perpendicular tangent can intersect the given tangent at any point on the parabola.
Therefore, the point on the line y = x^2 from which the other tangent to the parabola is perpendicular to the given tangent can be any point on the parabola.
The given tangent is y = x^2. To find the slope of this tangent, we take the derivative of y with respect to x:
dy/dx = 2x
This gives us the slope of the tangent at any point (x, x^2) on the parabola.
The slope of a perpendicular line is the negative reciprocal of the given slope. So, the slope of the perpendicular tangent is -1/(2x).
For the perpendicular tangent to be perpendicular to the given tangent, the product of their slopes should be -1:
(2x)*(-1/(2x)) = -1
Simplifying, we get:
-1 = -1
This is true for any value of x, so the perpendicular tangent can intersect the given tangent at any point on the parabola.
Therefore, the point on the line y = x^2 from which the other tangent to the parabola is perpendicular to the given tangent can be any point on the parabola.
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The equation of a tangent to the parabola y2 = 8x is y = x + 2. The point on this line from which the other tangent to the parabola is perpendicular to the given tangent is [2007]a)(2, 4)b)(–2, 0)c)(–1, 1)d)(0, 2)Correct answer is option 'B'. Can you explain this answer?
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