To find the value of 'a' for which the curves x^2 + ay^2 = 1 and y = x^2 intersect orthogonally, we need to determine the condition for orthogonal intersection and solve for 'a'.



When two curves intersect orthogonally, the tangent lines at the point of intersection are perpendicular to each other. Therefore, to find the value of 'a', we need to find the condition for perpendicular tangents.



The slopes of two perpendicular lines are negative reciprocals of each other. Hence, we can find the condition for perpendicular tangents by comparing the slopes of the tangent lines at the point of intersection.

Let's find the slopes of the tangent lines for each curve.



To find the slope of the tangent line, we differentiate the equation with respect to 'x'.

Differentiating both sides of the equation, we get:

2x + 2ayy' = 0
=> y' = -x/(ay)

The slope of the tangent line for curve 1 is given by y' = -x/(ay).



To find the slope of the tangent line, we differentiate the equation with respect to 'x'.

Differentiating both sides of the equation, we get:

y' = 2x

The slope of the tangent line for curve 2 is given by y' = 2x.



To compare the slopes and find the condition for perpendicular tangents, we equate the product of the slopes to -1.

(-x/(ay)) * (2x) = -1
=> -2x^2/(ay) = -1
=> x^2/(ay) = 1/2
=> x^2 = ay/2



To find the point of intersection, we substitute the equation of curve 2 into the equation x^2 = ay/2.

(x^2 = ay/2) and (y = x^2)

Substituting y = x^2 into the equation, we get:

x^2 = ax^2/2
=> 2x^2 - ax^2 = 0
=> x^2(2 - a) = 0

From this equation, we have two possibilities:

1. x^2 = 0
2. 2 - a = 0



If x^2 = 0, then x = 0. Substituting x = 0 into the equation y = x^2, we get y = 0.

Therefore, the point of intersection for x^2 + ay^2 = 1 and y = x^2 when x^2 = 0 is (0, 0).