Question -
Answer -
(i) focus is (3, 0) and the directrix is 3x + 4y = 1
Given:
The focus S(3, 0) and directrix(M) 3x + 4y – 1 = 0.
Let us assume P(x, y) be any point on the parabola.
The distance between two points (x1, y1)and (x2, y2) is given as:
And the perpendicular distance from the point (x1,y1) to the line ax + by + c = 0 is
So by equating both, we get
Upon cross multiplication, we get
25x2 + 25y2 – 150x + 225 =9x2 + 16y2 – 6x – 8y + 24xy + 1
16x2 + 9y2 – 24xy – 144x +8y + 224 = 0
∴The equation of the parabola is 16x2 + 9y2 –24xy – 144x + 8y + 224 = 0
(ii) focus is (1, 1) and the directrix is x + y + 1 = 0
Given:
The focus S(1, 1) and directrix(M) x + y + 1 = 0.
Let us assume P(x, y) be any point on the parabola.
The distance between two points (x1, y1)and (x2, y2) is given as:
And the perpendicular distance from the point (x1,y1) to the line ax + by + c = 0 is
So by equating both, we get
Upon cross multiplication, we get
2x2 + 2y2 – 4x – 4y + 4 = x2 +y2 + 2x + 2y + 2xy + 1
x2 + y2 + 2xy – 6x – 6y + 3= 0
∴ The equation of the parabola is x2 + y2 +2xy – 6x – 6y + 3 = 0
(iii) focus is (0, 0) and the directrix is 2x – y – 1 = 0
Given:
The focus S(0, 0) and directrix(M) 2x – y – 1 = 0.
Let us assume P(x, y) be any point on the parabola.
The distance between two points (x1, y1)and (x2, y2) is given as:
And the perpendicular distance from the point (x1,y1) to the line ax + by + c = 0 is
So by equating both, we get
Upon cross multiplication, we get
5x2 + 5y2 = 4x2 +y2 – 4x + 2y – 4xy + 1
x2 + 4y2 + 4xy + 4x – 2y – 1= 0
∴ The equation of the parabola is x2 + 4y2 +4xy + 4x – 2y – 1 = 0
(iv) focus is (2, 3) and the directrix is x – 4y + 1 = 0
Given:
The focus S(2, 3) and directrix(M) x – 4y + 3 = 0.
Let us assume P(x, y) be any point on the parabola.
The distance between two points (x1, y1)and (x2, y2) is given as:
And the perpendicular distance from the point (x1,y1) to the line ax + by + c = 0 is
So by equating both, we get
Upon cross multiplication, we get
17x2 + 17y2 – 68x – 102y +221 = x2 + 16y2 + 6x – 24y – 8xy + 9
16x2 + y2 + 8xy – 74x – 78y+ 212 = 0
∴ The equation of the parabola is 16x2 + y2 +8xy – 74x – 78y + 212 = 0