class 10th
chapter 9 (co-ordinate geometry)
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Step-by-step explanation:
The formula for distance between two points (x₁, y₁) and (x₂, y₂) is
√ [(x₂ - x₁)² + (y₂ - y₁)²]
So Distance between points (at₁², 2at₁) and (at₂², 2at₂) is
√ [(at₂² - at₁²)² + (2at₂ - 2at₁)²]
= √[a²(t₂² - t₁²)² + 4a²(t₂ - t₁)²]
//t₂² - t₁² is of form a² - b², thus t₂² - t₁² = (t₂ - t₁)(t₂ + t₁)
= √ [a²{(t₂ - t₁)(t₂ + t₁)}² + 4a²(t₂ - t₁)²]
= √ [a²{(t₂ - t₁)²(t₂ + t₁)²} + + 4a²(t₂ - t₁)²]
//a²(t₂ - t₁)² is common
= √ [ a²(t₂ - t₁)² {(t₂ + t₁)² + 4} ]
//Bring out a²(t₂ - t₁)² out of the square root which becomes a(t₂ - t₁)
= a(t₂ - t₁) √[(t₂ + t₁)² + 4]
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