If the line x+4 y= 7 , where R , touches the ellipse 3 x^ 2 +4 y^ 2 =1 at the point P in the first quadrant, then one of the focal distances of P is :

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Correct answer: A. 1 3 + 1 2 7

Ellipse 3x^2 + 4y^2 = 1: a^2 = 1 3 , b^2 = 1 4 , e = 1 2 . Tangent at (x_0, y_0): 3x_0x + 4y_0y = 1. Comparing with x + 4y = 7 : 4y_0 = 4 7 y_0 = 1 7 and 3x_0 = 7 . Substituting in ellipse equation: ^2 21 + 4 7 = 1 ^2 = 9 = 3. P = ( 1 7 , 1 7 ). Focal distances = a ex_0 = 1 3 1 2 7 . One focal distance is 1 3 + 1 2 7 .

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