The line through (P(a, 2) ), where (a 0 ), making an angle (45^ ) with the positive direction of the (X )-axis meets the curve ( x^2 9 + y^2 4 =1 ) at (A ) and (D ) and the coordinate axes at (B ) and (C ). If (P A, P B, P C ) and (P D ) are in a geometric progression, then (2 a= )

Step-by-step solution

Since equation of line passes through (P(a, 2) ), (a 0 ) making an angle (45^ ) with positive direction of the (X )-axis is : ( x-a 1 2 = y-2 1 2 =r x=a+ r 2 and y=2+ r 2 ) So for point (B ) (as it is on (X )-axis), so, (r=-2 2 ) so (B(a-2,0) ) for point (C ) (as it is on (Y )-axis), so (r=-a 2 ) so, (C(0,2-a) ) ( aligned P B & = 4+4 =2 2 P C & = a^2+a^2 =a 2 aligned ) For points (A ) and (D ) ( gathered (a+ r 2 )^2 9 + (2+ r 2 )^2 4 =1 4 (a+ r 2 )^2+9 (2+ r 2 )^2=36 13 ( r^2 2 )+ ( 8 a 2 + 36 2 ) r+4 a^2=0 , let h