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Find the equation of the normal at a point on the curve x2 = 4y which passes through the point (1.2)
- a)y = x - 1
- b)2x + y = 4
- c)x-y = 3
- d)x+y = 3
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Find the equation of the normal at a point on the curve x2 = 4ywhich p...
The equation of the given curve is x2 = 4y
Differentiating w.r.t. x, we get 
Now, slope of the tangent at (h,k) is given by, 
Thus, slope of the normal at 
Therefore, the equation of normal at
Since it passes through the point (1,2) we have
now(h,k) lies on the curve x2 = 4y, so we have h2 = 4k ...(3)
Solving (2) and (3), we get ℎ=2 and k= 1
From (1), the required equation of the normal is:
Since it passes through the point (1,2) we have
now(h,k) lies on the curve x2 = 4y, so we have h2 = 4k ...(3)
Solving (2) and (3), we get ℎ=2 and k= 1
From (1), the required equation of the normal is:
Most Upvoted Answer
Find the equation of the normal at a point on the curve x2 = 4ywhich p...
Understanding the Curve
The given curve is x^2 = 4y, which is a parabola that opens upwards. To find the equation of the normal at a point (x₁, y₁) on this curve, we first need to determine the coordinates of the point on the curve.
Finding the Point on the Curve
1. Substituting y: From the equation x^2 = 4y, we can express y as:
- y = (x^2)/4
2. Point of Interest: We need to find the point on the curve where the normal passes through (1, 2). Let’s denote the point on the curve as (x₁, y₁) = (x₁, x₁^2/4).
Finding the Slope of the Tangent
1. Derivative: To find the slope of the tangent line at (x₁, y₁), we differentiate:
- dy/dx = x/2 (Differentiating x^2 = 4y).
2. Slope of Normal: The slope of the normal line is the negative reciprocal of the tangent slope:
- Slope of normal = -2/x₁.
Equation of the Normal
1. Using Point-Slope Form: The equation of the normal can be expressed as:
- y - y₁ = (-2/x₁)(x - x₁).
2. Substituting y₁: Replace y₁ with (x₁^2)/4:
- y - (x₁^2)/4 = (-2/x₁)(x - x₁).
Finding x₁ for Normal Passing through (1, 2)
1. Substituting Point: For the normal to pass through (1, 2):
- 2 - (x₁^2)/4 = (-2/x₁)(1 - x₁).
2. Solving Equation: After solving, you will find possible values for x₁.
Final Equation of Normal
After determining x₁, you will find the specific equation of the normal. The correct answer among the options is option 'D': x + y = 3.
This equation is derived based on the calculations showing that the normal to the curve at the determined point intersects (1, 2).
The given curve is x^2 = 4y, which is a parabola that opens upwards. To find the equation of the normal at a point (x₁, y₁) on this curve, we first need to determine the coordinates of the point on the curve.
Finding the Point on the Curve
1. Substituting y: From the equation x^2 = 4y, we can express y as:
- y = (x^2)/4
2. Point of Interest: We need to find the point on the curve where the normal passes through (1, 2). Let’s denote the point on the curve as (x₁, y₁) = (x₁, x₁^2/4).
Finding the Slope of the Tangent
1. Derivative: To find the slope of the tangent line at (x₁, y₁), we differentiate:
- dy/dx = x/2 (Differentiating x^2 = 4y).
2. Slope of Normal: The slope of the normal line is the negative reciprocal of the tangent slope:
- Slope of normal = -2/x₁.
Equation of the Normal
1. Using Point-Slope Form: The equation of the normal can be expressed as:
- y - y₁ = (-2/x₁)(x - x₁).
2. Substituting y₁: Replace y₁ with (x₁^2)/4:
- y - (x₁^2)/4 = (-2/x₁)(x - x₁).
Finding x₁ for Normal Passing through (1, 2)
1. Substituting Point: For the normal to pass through (1, 2):
- 2 - (x₁^2)/4 = (-2/x₁)(1 - x₁).
2. Solving Equation: After solving, you will find possible values for x₁.
Final Equation of Normal
After determining x₁, you will find the specific equation of the normal. The correct answer among the options is option 'D': x + y = 3.
This equation is derived based on the calculations showing that the normal to the curve at the determined point intersects (1, 2).
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Find the equation of the normal at a point on the curve x2 = 4ywhich passes through the point (1.2)a)y = x - 1b)2x + y = 4c)x-y = 3d)x+y = 3Correct answer is option 'D'. Can you explain this answer?
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