Let E be the locus of a point M such that the ratio of its distance from the point S(3,0) to its perpendicular distance from the line 3x - 25 = 0 is 3 : 5 . A tangent to E in the first quadrant is parallel to the straight line 3x + 5y = 10 . The area of the triangle formed by this tangent and the coordinate axes is

Step-by-step solution

Let the coordinates of M be (x, y) . The given condition states that the ratio of the distance of M from S(3,0) to its distance from the line 3x - 25 = 0 is 3 5 . This represents the focus-directrix definition of an ellipse, where the focus is S(3,0) , the directrix is x = 25 3 , and the eccentricity is e = 3 5 . For a standard ellipse x^2 a^2 + y^2 b^2 = 1 , the focus is at (ae, 0) and the directrix is x = a e . Thus, ae = 3 and a e = 25 3 . Multiplying these relations gives a^2 = 25 a = 5 . Using b^2 = a^2(1 - e^