Let C be the circle x2 + (y - 1)2 = 2, E1 and E2 be two ellipses whose centres lie at the origin and major axes lie on x-axis and y-axis respectively. Let the straight line x + y = 3 touch the curves C, E1 and E2 at P(x1, y1), Q(x2, y2) and R(x3, y3) respectively. Given that P is the mid point of the line segment QR and PQ = 2 23, the value of 9(x1y1 + x2y2 + x3y3) is equal to .
Answer
46
Solution
Circle C: centre (0, 1), radius 2. Distance of centre from x + y = 3 is |0 + 1 - 3| 2 = 2, so the line is a tangent.
P = foot of the perpendicular from (0, 1): (t, 1 + t) on the line 2t + 1 = 3 t = 1
P = (1, 2) x1y1 = 2
Unit vector along the line: 1 2(1, -1). Q and R are at distance 2 23 on either side of P:
P 2 23 1 2(1, -1) = (1 23, 2 23 )
the two points are ( 53, 43 ) and ( 13, 83 )
For x2a2 + y2b2 = 1, the tangent x + y = 3 touches at ( a23, b23 ) with a2 + b2 = 9.
E1 (major axis on x-axis, a > b) needs x > y at the point of contact Q = ( 53, 43 ) (a2 = 5, b2 = 4)
E2 (major axis on y-axis) R = ( 13, 83 ) (a2 = 1, b2 = 8)
x2y2 = 209, x3y3 = 89
9(x1y1 + x2y2 + x3y3) = 9 (2 + 209 + 89 )
= 18 + 20 + 8 = 46
Answer: 46