Find the Equation of parabola whose focus is [3,5] and vertex is [1,3]?
Answers
Given : parabola whose focus is S(3,5) and vertex is A(1,3).
To Find : the equation of the parabola
Solution:
Vertex is A ( 1 , 3)
Focus if ( 3 , 5)
Equation of axis
y - 3 = {( 5 - 3)/(3 - 1) } (x - 1)
=> y - 3 = x - 1
=> y = x + 2
slope of axis = 1
Hence slope of directrix = -1
y = - x + c
x = h then y = -h + c
point ( h , -h + c) is on the directrix lying on axis
(h + 3)/2 = 1 ( -h + c + 5)/2 = 3
=> h = - 1 , => c = 0 using h = - 1
x= - 1 , y = 1
x + y = 0 is directrix
point ( x , y ) on parabola
Distance from directrix = distance from focus
√x - 3)² + ( y - 5)² = | (x + y)/(√1² + 1²) |
Squaring both sides
=> x² -6x + 9 + y² -10y + 25 = ( x² + y² + 2xy)/2
=> x² + y² - 2xy -12x - 20y + 68 = 0
(-2)² - (4)(1)(1) = 0 Hence Parabola
Additional Info : To understand how its parabola
Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0
D = B^2 - 4AC
having xy-term, rotates the graph and its shape, .
The discriminant (B2-4AC) is used to determine which conic section will result.
If the discriminant is less than zero we have a circle (if A = C) or an ellipse;
if the discriminant is equal to zero we have a parabola;
if the discriminant is greater than zero we have a hyperbola.
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