The point (x 1 , y 1 ) lies outside, on or inside the parabola y² = 4ax according as the expression S 1 = y 1 ² − 4ax 1 is positive, zero or negative.
The point P(x 1, y 1 ) lies outside, inside or on the ellipse according as S 1 = 
positive, negative or zero.
The point P(x 1, y 1 ) lies outside, inside or on the hyperbola according as S 1 = 
– 1 negative , positive or zero.
(i) The position of the point (5, – 4) relative to the hyperbola y 2 –9x 2 + 1 = 0
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
(i) Since 9 (5) 2 – (–4) 2 – 1 = 225 – 16 – 1 = 208 > 0,
So the point (5,–4) lies inside the hyperbola 9x 2 – y 2 = 1.
(ii) x 2 + x + 1 – y = 0
( α , 2 α + 1)
α 2 + α + 1 – (2 α + 1) ≤ 0
⇒ α 2 – α ≤ 0 ⇒ α ( α – 1) ≤ 0 ⇒ α∈ [0, 1]
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