Question

Prosta y=x+4 przecina okrąg x² + y² =25 w punktach A i B. Oblicz cięciwę AB

Answer 1

y = x + 4

x² + y² = 25

 

x² + ( x + 4 )² = 25

x² + x² + 8x + 16 - 25 = 0

2x² + 8x - 9 = 0

Δ = b² - 4ac

Δ = 64 + 72 = 136

√Δ = 2√34

x₁= ( - 8 - 2√34) : 4 = - 2 - 1/2√34                  y₁ = - 2 - 1/2√34 + 4 = 2 - 1/2√34

x₂ = - 2 + 1/2√34                                            y₂ = - 2 + 1/2√34  + 4 = 2 + 1/2√34

 

A = ( - 2 - 1/2√34;  2 - 1/2√34)      B = ( - 2 + 1/2√34 ;  2 + 1/2√34 )

I ABI² = (x₂ - x₁ )² + (y₂ - y₁)²

IABI² =(  - 2 + 1/2√34 + 2 + 1/2√34)² + (  2 + 1/2√34 - 2 + 1/2√34)² = ( √34 )² + ( √34 )² = 34 + 34 = 68

IABI = √68 = 2√17