Step-by-step explanation:
Given point are: P(at², 2at)
Q(a/t², 2a/t)
S(0, 0)
Now, the distance between A(x₁y₁) and B(X₂, y₂)
then AB = √{(x₂ - x₁)² + (y₂ - y₁)²} units
(i) The distance between S and P:
(x₁, y₁) = (0, 0) ⇛x₁= 0, y₁ = 0
(x₂, y₂ ) = (at², 2at) ⇛x₂ = at², y₂ = 2at
SP = √{at² - 0)² + (2at - 0)²}
= √{(at²)² + (2at)²}
= √{a²t²*² + 4a²t²}
= √{a²t⁴ + 4a²t²}
= √{a²t²(t²+4)}
SP = at√(t² + 4)→→→Eqn(1)
(ii) The distance between S and Q :
(x₁, y₁) = (0, 0) ⇛x₁= 0, y₁ = 0
(x₂, y₂ ) = (a/t², 2a/t) ⇛x₂ = a/t², y₂ = 2a/t
SQ = √[{(a/t²) - 0} + {(2a/t) - 0}
= √{(a/t²)² + (2a/t)²}
= √{(a²/t²*²) + (4a²/t²)}
= √{(a²/t⁴) + (4a²/t²)}
= √{(a² + 4a²t²)/t⁴}
= √[{a²(1 + 4t²)}/t⁴]
SQ = (a/t²)√(1 + 4t²) →→→ Eqn(2)
Now,
(1/SP) + (1/SQ) = [1/{at√(t² + 4)}] + [1/{(a/t²)√(1 + 4t²)}]
= (1/at)[1/{√(t² + 4)}] + (t²/a)[1/{(√1 + 4t²)}]
= (1/a)[[1/{t√(t² + 4)}] + [t²/{√(1 + 4t²)}]]
(1/SP) + (1/SQ) = 1/a is not independent of 't'
If suppose S = (a, 0) then
SP = √{(at² - a)² + (2at - 0)²}
= √{a²(t² - 1)² + (2at)²}
= a√{(t² - 1)² + 4t²}
= a√{(t² + 1)²}
SP = a(t² + 1)
1/SP = 1/{a(t² + 1)} →→→Eqn(1)
And
SQ = √[{(a/t²) - a}² + {(2a/t) - 0}²]
= √[a²{(1/t²) - 1}² + a²(2/t)²]
= a√[{(1 - t²)²/t⁴} + (4/t²)]
= a√[{(1 - t²)² + 4t²}/t⁴]
( a/t²)√(1 + t²)²
SQ = (a/t²)(1 + t²)
1/SQ = 1/{(a/t²)(1 + t²)} = t²/{a(1 + t²)} →→→Eqn(2)
Therefore, (1/SP) + (1/SQ)
= 1/{a(t² + 1)} + t²/{a(1 + t²)}
= (1 + t²)/a(1 + t²)
= 1/a
1/a is independent of 't'.
