Let P and Q be points of the ellipse `16 x^(2) +25y^(2) = 400` so that `PQ = 96//25` and P and Q lie above major axis. Circle drawn with PQ as diameter touch major axis at positive focus, then the value of slope m of PQ is

Options

A. `-1`
B. `1//2`
C. 2
D. `1//3`

Answer

A

Explanation

Ellipse `16x^(2) + 25y^(2) = 400`
`rArr (x^(2))/(25) + (y^(2))/(16) =1`
Let the points on ellipse be `P(5 cos theta_(1),4 sin theta_(1))` and `Q(5 cos theta_(2), 4 sin theta_(2))`
Circle described on PQ as diameter touches x-axis (3,0)
`5((cos theta_(1)+cos theta_(2))/(2))=3` and `4 ((sin theta_(1)+sin theta_(2))/(2)) =r`
`rArr (4)/(5) tan ((theta_(1)+theta_(2))/(2)) =(r)/(3) rArr tan ((theta_(1)+theta_(2))/(2)) =(5r)/(12)`
`rArr` Slope of `PQ =- (4)/(5) cot. (theta_(1)+theta_(2))/(2) =- (48)/(25r) =-1`