Understanding the Problem:

In the context of calculus and geometry, we often deal with parabolic curves and the tangents to these curves. Sometimes, the tangents can intersect at a specific point, like the origin. This article explores the process of finding the angle between the tangents to a particular parabolic curve, which is defined by the equation x 3 (frac{1}{4}y^2).

Step-by-Step Solution:

To solve this problem, we will follow a systematic approach:

1. Rewriting the Curve in Standard Form:

The given curve equation is:

x 3 (frac{1}{4}y^2)

By rearranging the equation, we get:

x - 3 (frac{1}{4}y^2)

This equation represents a parabola that opens to the right.

2. Finding the Derivatives:

To find the slopes of the tangents, we implicitly differentiate the given equation:

[frac{dx}{dy} frac{1}{2}y^2]

From this, we can determine the slope of the tangent line at any point on the curve:

[frac{dy}{dx} frac{2}{y^2}]

3. Determining the Points of Tangency:

Let the point of tangency be ((x_0, y_0)). The tangent lines at this point will have the slope:

[m frac{2}{y_0^2}]

The equation of the tangent line at this point can be written in point-slope form:

[y - y_0 m(x - x_0)]

Rearranging this yields:

[y mx - mx_0 y_0]

4. Finding theIntersection with the Origin:

For the tangent lines to intersect at the origin (0, 0), we set:

[0 m(0) - mx_0 y_0]

This simplifies to:

[y_0 mx_0]

Substituting (y_0 mx_0) back into the parabola equation:

[x_0 3 frac{1}{4}(mx_0)^2]

From this equation, we can solve for (x_0) and (m). This will provide the points of tangency.

5. Finding the Angles Between Tangents:

Once we have the slopes (m_1) and (m_2), we can use the formula for the angle between two lines:

[tan theta left|frac{m_1 - m_2}{1 m_1 m_2}right|]

This will allow us to calculate the angle (theta) between the two tangent lines that intersect at the origin.

Final Calculation:

After solving the quadratic equation, we can substitute the obtained slopes into the angle formula to compute the angle (theta).

In conclusion, by following these steps, we can determine the angle between the tangents to the parabola defined by (x 3 frac{1}{4}y^2) at the origin. If you need further assistance or clarification on any part of this process, feel free to ask!