Let P (x_1, y_1 ) and Q (x_2, y_2 ) be two distinct points on the ellipse x^2 9 + y^2 4 =1 such that y_1>0, and y_2>0. Let C denote the circle x^2+y^2=9, and M be the point (3,0). Suppose the line x=x_1 intersects C at R, and the line x=x_2 intersects C at S, such that the y-coordinates of R and S are positive. Let R O M= 6 and S O M= 3 , where O denotes the origin (0,0). Let |X Y| denote the length of the line segment X Y. Then which of the following statements is (are) TRUE?
C. If N_2= (x_2, 0 ), then 3 |N_2 Q |=2 |N_2 S |
aligned & P (3 30^ , 2 30^ ) ( 3 3 2 , 1 ) \\ & Q (3 60^ , 2 60^ ) ( 3 2 , 3 ) \\ & R ( 3 3 2 , 3 2 ), S ( 3 2 , 3 3 2 ) aligned Slope of PQ = m _ PQ = 3 -1 3 2 - 3 3 2 =- 2 3 Equation of line PQ y - 3 =- 2 3 ( x - 3 2 ) 2 x+3 y=3( 3 +1) option (A) is correct Now aligned & if N _2= ( x _2, 0 )= ( 3 2 , 0 ) \\ & | N _2 Q |= 3 and | N _2 ~S |= 3 3 2 aligned 3 | ~N _2 Q |=2 | ~N _2 ~S | option (C) is correct Now, if N _1= ( x _1, 0 ) N _1= ( 3 3 2 , 0 ) | N _1 P |=1, | ~N _1 R |= 3 2 option (D) is incorrect
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