2. Find a family of curves such that each of its members intersects every parabola in the family y-1=Cx^2 at right angles.

Here's how to find the orthogonal family of curves:1. **Find the differential equation representing the given family of parabolas:**The given family is `y - 1 = Cx²`. We need to eliminate the arbitrary constant 'C'. Differentiating both sides with respect to 'x', we get:dy/dx = 2CxFrom the original equation, C = (y - 1)/x². Substituting this into the derivative:dy/dx = 2[(y - 1)/x²] * xdy/dx = 2(y - 1)/xThis is the differential equation representing the given family of parabolas.2. **Find the differential equation of the orthogonal family:**For two curves to be orthogonal (intersect at right angles), the product of their slopes at the point of intersection must be -1. Let the slope of the orthogonal family be dy'/dx. Then:(dy/dx) * (dy'/dx) = -1Substitute the previously found dy/dx:[2(y - 1)/x] * (dy'/dx) = -1dy'/dx = -x / [2(y - 1)]This is the differential equation representing the orthogonal family of curves.3. **Solve the differential equation to find the orthogonal family:**This is a separable differential equation. Rearranging:2(y - 1) dy' = -x dxIntegrating both sides:∫ 2(y - 1) dy = ∫ -x dx(y - 1)² = -x²/2 + k (where 'k' is the constant of integration)Therefore, the family of curves orthogonal to the given parabolas is:(y - 1)² = -x²/2 + kor equivalentlyx² + 2(y - 1)² = 2kSince 'k' is an arbitrary constant, we can replace 2k with another constant, say 'c':x² + 2(y - 1)² = cThis represents a family of ellipses centered at (0, 1).Therefore, the family of ellipses x² + 2(y - 1)² = c intersects each parabola of the family y - 1 = Cx² at right angles.