equation of the pa
Find the equation of the parabola whose focus is S(3,5) and vertex is A(1,3).
Find the equation of the parabola whose latus rectum is the line segment joining
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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