the equation of the parabola with a directrix of y=2 and a focus of (8,6)

Answer 1

IN PRABOLA THE DISTANCE BETWEEN ANY POINT (X,Y) IN THE CURVE AND THE FOCUS (a,b), HAS TO BE EQUAL THE DISTANCE BETWEEN THIS POINT (X,Y) AND THE DIRECTRIX y= k

√ (y - k )² = √ ( x -a)² + ( y- b)²

square both sides,

(y -k )² = ( x - a)² + ( y- b)²

y²- 2yk +k² = ( x - a)² + y² - 2by + b²

- 2yk +k² = ( x - a)² - 2by + b²

2(b-k) y = ( x - a)² + b² - k²

y = ( x - a)² / 2(b-k) + ( b² - k² ) / 2(b-k)

y = ( x - a)² / 2(b-k) + ( b +k ) ( b-k) / 2(b-k)

y= ( x - a)² / 2(b-k) + ( b+ k ) /2

substitute k=2, a=8 ,and b= 6

y =( x - 8)² / 2(6-2) + ( 6+ 2 ) /2

y =( x - 8)² / 8 + 4