Question

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.

Answer 1

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 }

To Know more:

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