Passage: Consider the ellipses given by x^2 + 4y^2 = 1 and 4x^2 + y^2 = 1 . Question: Let P be the point in the first quadrant where the given ellipses intersect. If is the acute angle between the tangents to the given ellipses at the point P , then the value of 4 is ___________.

Step-by-step solution

The given equations of the ellipses are x^2 + 4y^2 = 1 and 4x^2 + y^2 = 1 . To find the point of intersection, we equate the two expressions: x^2 + 4y^2 = 4x^2 + y^2 3x^2 = 3y^2 x = y (since P lies in the first quadrant) Substituting x = y into the first ellipse equation: x^2 + 4x^2 = 1 5x^2 = 1 x = 1 5 Thus, the point of intersection is P ( 1 5 , 1 5 ) . Differentiating x^2 + 4y^2 = 1 with respect to x gives the slope of the tangent to the first ellipse: 2x + 8y dy dx = 0 dy dx = - x 4y At P ( 1 5 , 1 5 ) , the sl