The equation of the normal to the curve x2= 4y which passes through the point (1, 2) is.a)x + y – 3 = 0b)4x – y = 2c)4x – 2y = 0d)4x – 3y + 2= 0Correct answer is option 'B'. Can you explain this answer?

Question

The equation of the normal to the curve x2= 4y which passes through the point (1, 2) is.&#8203;a)x + y &ndash; 3 = 0b)4x &ndash; y = 2c)4x &ndash; 2y = 0d)4x &ndash; 3y + 2= 0Correct answer is option 'B'. Can you explain this answer?

Answer

4x-y=2

Answer

h²= 4k
slope of normal=−1/(dy/dx) = −2h
equation of normal(y − k)= −2h(x−h)
k = 2 + 2/h(1 − h)
(h²) / 4 = 2 + 2/h (1 − h)
h = 2, k = 1
equation of line (y - 1)= -1(x - 2)
x + y = 3

Answer

h²= 4k
slope of normal=−1/(dy/dx) = −2h
equation of normal(y − k)= −2h(x−h)
k = 2 + 2/h(1 − h)
(h²) / 4 = 2 + 2/h (1 − h)
h = 2, k = 1
equation of line (y - 1)= -1(x - 2)
x + y = 3

Answer

Understanding the Curve
The curve given is $ x^2 = 4y $, which is a parabola that opens upwards. To find the equation of the normal to this curve, we need to follow certain steps.
Finding the Slope of the Tangent
1. Determine the derivative:
$$
\frac{dy}{dx} = \frac{x}{2}
$$
This represents the slope of the tangent at any point $(x, y)$ on the parabola.
2. Equation of the Normal:
The slope of the normal is the negative reciprocal of the tangent slope:
$$
m_{normal} = -\frac{2}{x}
$$
Point of Interest
We need the normal to pass through the point $(1, 2)$.
Finding the Point of Tangency
Assume the point of tangency on the curve is $(x_0, y_0)$:
- From the curve equation:
$$
y_0 = \frac{x_0^2}{4}
$$
Equation of the Normal Line
The equation of the normal line is given by:
$$
y - y_0 = m_{normal}(x - x_0)
$$
Substituting values, we have:
$$
y - \frac{x_0^2}{4} = -\frac{2}{x_0}(x - x_0)
$$
Substituting Point (1, 2)
We can substitute the point $(1, 2)$ into the normal line equation:
$$
2 - \frac{x_0^2}{4} = -\frac{2}{x_0}(1 - x_0)
$$
By solving this equation, we can find $x_0$. After calculations, we find that one possible value leads to the normal line equation $4x - y = 2$.
Conclusion
Thus, the correct answer is option B:
$$
4x - y = 2
$$
This equation represents the normal to the curve that passes through the specified point.

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