Condition for a line y=mx+c tobe a normal to a parabola y^2=4ax is c=-2am-am^3.Derive it!

To understand the condition for a line represented by the equation y = mx + c to be a normal to the parabola given by y² = 4ax, let’s break down the concepts and derive the condition step by step.

Understanding the Parabola

The equation y² = 4ax describes a standard parabola that opens to the right. Here, 'a' is a constant that determines the width of the parabola. The vertex of this parabola is at the origin (0, 0), and any point on the parabola can be expressed in parametric form as:

• x = at²
• y = 2at

The Slope of the Tangent Line

For any point (at², 2at) on the parabola, we can find the slope of the tangent line using differentiation. The derivative of y² = 4ax with respect to x gives:

2y(dy/dx) = 4a

Thus, the slope (dy/dx) of the tangent line at any point on the parabola is:

dy/dx = 2a/y = 2a/(2at) = 1/t.

Finding the Slope of the Normal

The slope of the normal line at the point (at², 2at) is the negative reciprocal of the slope of the tangent. Therefore, the slope of the normal line is:

m_normal = -t.

Condition for the Normal Line

Now, we need to express the line y = mx + c in terms of its relation to the parabola. We have established that for the normal line, m = -t. Hence, we can substitute:

y = -tx + c.

At the point of tangency (at², 2at), the normal line should pass through this point. Therefore, substituting x = at² and y = 2at into the normal line equation gives:

2at = -t(at²) + c.

Rearranging this equation leads to:

c = 2at + at³.

Relating c to m

Since we have m = -t, we can express t in terms of m:

t = -m.

Now, substituting t back into the equation for c results in:

c = 2a(-m) + a(-m)³ = -2am - am³.

Final Condition

Thus, we find that the condition for the line y = mx + c to be a normal to the parabola y² = 4ax is:

c = -2am - am³.

By deriving this condition, we've established a clear relationship between the parameters of the line and the characteristics of the parabola. This analysis can be useful in various applications, from geometry to physics, where understanding the properties of curves is essential.