Answer is: option4

\( y = -\frac{1}{2} \ln x \)

Solution:

Finding the Equation of Orthogonal Trajectories

To find the equation of a curve that intersects every curve in the family \( y = x^2 + c \) at right angles, we need to determine the orthogonal trajectories.

The given family of curves is:

\[ y = x^2 + c \]

Differentiating both sides with respect to \( x \):

\[ \frac{dy}{dx} = 2x \]

Two curves intersect orthogonally if the product of their slopes at the point of intersection is \( -1 \). Let \( m \) be the slope of the required curve. Then:

\[ m \cdot 2x = -1 \]

Thus, the required slope is:

\[ m = -\frac{1}{2x} \]

Equation for the Orthogonal Trajectories

The equation for the orthogonal trajectories is:

\[ \frac{dy}{dx} = -\frac{1}{2x} \]

Separating variables:

\[ dy = -\frac{1}{2x} dx \]

Integrating both sides:

\[ y = -\frac{1}{2} \ln |x| + C \]

From the given choices, the equation that matches our result is:

\[ y = -\frac{1}{2} \ln x \]

which corresponds to option (D).