The line x = 8 is the directrix of the ellipse E : x 2 a 2 + y 2 b 2 = 1 with the corresponding focus 2 , 0 . If the tangent to E at the point P in the first quadrant passes through the point 0 , 4 3 and intersects the x -axis at Q , then 3 P Q 2 is equal to _____ .

Step-by-step solution

Given, The line x = 8 is the directrix of the ellipse E : x 2 a 2 + y 2 b 2 = 1 with the corresponding focus 2 , 0 , So, equation of directrix is given by x = a e ⇒ 8 = a e             … 1 And distance of corresponding focus is given by, a e = 2                     … 2 So, from both equation we get, 8 e = 2 e ⇒ e 2 = 1 4 ⇒ e = 1 2 Hence, a = 4 Now using eccentricity formula we get, b 2 = a 2 1 - e 2 = 16 3