6. Find the equation of a parabola whose focus is the point (1, 1) and whose directrix is the line 3 x + 4y - 2 = 0. Also, find the equation of its axis and the coordinates of its vertex.
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Equation of parabola
Step-by-step explanation:
- Axis and directrix of parabola are perpendicular to each other.
- Directrix equation is given as 3x+4y-2 = 0, axis equation is of the form 4x-3y+c = 0, where c is a constant.
- Since axis is passing through the focus S(1,1), substitute x=1, y=1 in axis equation and find constant c.
4x-3y+c = 4(1) - 3(1) +c= 0,
hence c = -1 ;
Equation of axis is 4x-3y-1 = 0
- Intersection point P of axis and directrix is obtained by solving both directrix and axis equations as shown below
3x+4y = 2 .....................(1)
4x-3y = 1 .....................(2)
- By solving the linear equations(1) and (2) , we get x = 2/5, y = 1/5 ;
- Hence coordinates of point of intersection P is (2/5 , 1/5)
- Hence vertex point is { [1+(2/5)]/2, [1+1/5)]/2 } = { 7/10, 6/10 }
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