Consider the two curves C1 : y2 = 4x ; C2 : x2 + y2 – 6x + 1, then
C1 and C2 touch each other exactly at two points
Solving the two equations, we get
x2 + 4x – 6x + 1 = 0
⇒ x2 + 2x + 1 = 0
⇒ x = 1 and y = ±2
So the two curve meet at two points (1, 2) and (1, –2). Equation of tangents at (1, 2) to C1is y(2) = 2 (x + 1)
⇒ y = x +1
and equation of the tangent at (1, 2) to C2 is
x. 1 + y(2) – 3(x + 1) + 1 = 0 ⇒ y = x + 1
Showing that C1 and C2 have a common tangent at the point (1, 2).
Similarly they have a common tangent
y = –(x + 1) at (1, –2)
Hence the two curve touch each other exactly at two points.
If F1 = (3, 0), F2 = (–3, 0) and P is any point on the curve 16x2 +25y2 = 400, then PF1 + PF2 equal
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