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25. Parabola
Exercise 25.1
1 A. Question
Find the equation of the parabola whose:
focus is (3, 0) and the directrix is 3x + 4y = 1
Answer
Given that we need to find the equation of the parabola whose focus is S(3, 0) and directrix(M) is 3x + 4y - 1
= 0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 25x
2
 + 25y
2
 - 150x + 225 = 9x
2
 + 16y
2
 - 6x - 8y + 24xy + 1
? 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
?The equation of the parabola is 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
1 B. Question
Find the equation of the parabola whose:
focus is (1, 1) and the directrix is x + y + 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(1, 1) and directrix(M) is x + y + 1 =
0.
Page 2


25. Parabola
Exercise 25.1
1 A. Question
Find the equation of the parabola whose:
focus is (3, 0) and the directrix is 3x + 4y = 1
Answer
Given that we need to find the equation of the parabola whose focus is S(3, 0) and directrix(M) is 3x + 4y - 1
= 0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 25x
2
 + 25y
2
 - 150x + 225 = 9x
2
 + 16y
2
 - 6x - 8y + 24xy + 1
? 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
?The equation of the parabola is 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
1 B. Question
Find the equation of the parabola whose:
focus is (1, 1) and the directrix is x + y + 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(1, 1) and directrix(M) is x + y + 1 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 2x
2
 + 2y
2
 - 4x - 4y + 4 = x
2
 + y
2
 + 2x + 2y + 2xy + 1
? x
2
 + y
2
 + 2xy - 6x - 6y + 3 = 0
?The equation of the parabola is x
2
 + y
2
 + 2xy - 6x - 6y + 3 = 0.
1 C. Question
Find the equation of the parabola whose:
focus is (0, 0) and the directrix is 2x - y – 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(0, 0) and directrix(M) is 2x - y - 1 =
0.
Page 3


25. Parabola
Exercise 25.1
1 A. Question
Find the equation of the parabola whose:
focus is (3, 0) and the directrix is 3x + 4y = 1
Answer
Given that we need to find the equation of the parabola whose focus is S(3, 0) and directrix(M) is 3x + 4y - 1
= 0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 25x
2
 + 25y
2
 - 150x + 225 = 9x
2
 + 16y
2
 - 6x - 8y + 24xy + 1
? 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
?The equation of the parabola is 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
1 B. Question
Find the equation of the parabola whose:
focus is (1, 1) and the directrix is x + y + 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(1, 1) and directrix(M) is x + y + 1 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 2x
2
 + 2y
2
 - 4x - 4y + 4 = x
2
 + y
2
 + 2x + 2y + 2xy + 1
? x
2
 + y
2
 + 2xy - 6x - 6y + 3 = 0
?The equation of the parabola is x
2
 + y
2
 + 2xy - 6x - 6y + 3 = 0.
1 C. Question
Find the equation of the parabola whose:
focus is (0, 0) and the directrix is 2x - y – 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(0, 0) and directrix(M) is 2x - y - 1 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 5x
2
 + 5y
2
 = 4x
2
 + y
2
 - 4x + 2y - 4xy + 1
? x
2
 + 4y
2
 + 4xy + 4x - 2y - 1 = 0
?The equation of the parabola is x
2
 + 4y
2
 + 4xy + 4x - 2y - 1 = 0.
1 D. Question
Find the equation of the parabola whose:
focus is (2, 3) and the directrix is x - 4y + 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(2, 3) and directrix(M) is x - 4y + 3 =
0.
Page 4


25. Parabola
Exercise 25.1
1 A. Question
Find the equation of the parabola whose:
focus is (3, 0) and the directrix is 3x + 4y = 1
Answer
Given that we need to find the equation of the parabola whose focus is S(3, 0) and directrix(M) is 3x + 4y - 1
= 0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 25x
2
 + 25y
2
 - 150x + 225 = 9x
2
 + 16y
2
 - 6x - 8y + 24xy + 1
? 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
?The equation of the parabola is 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
1 B. Question
Find the equation of the parabola whose:
focus is (1, 1) and the directrix is x + y + 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(1, 1) and directrix(M) is x + y + 1 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 2x
2
 + 2y
2
 - 4x - 4y + 4 = x
2
 + y
2
 + 2x + 2y + 2xy + 1
? x
2
 + y
2
 + 2xy - 6x - 6y + 3 = 0
?The equation of the parabola is x
2
 + y
2
 + 2xy - 6x - 6y + 3 = 0.
1 C. Question
Find the equation of the parabola whose:
focus is (0, 0) and the directrix is 2x - y – 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(0, 0) and directrix(M) is 2x - y - 1 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 5x
2
 + 5y
2
 = 4x
2
 + y
2
 - 4x + 2y - 4xy + 1
? x
2
 + 4y
2
 + 4xy + 4x - 2y - 1 = 0
?The equation of the parabola is x
2
 + 4y
2
 + 4xy + 4x - 2y - 1 = 0.
1 D. Question
Find the equation of the parabola whose:
focus is (2, 3) and the directrix is x - 4y + 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(2, 3) and directrix(M) is x - 4y + 3 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 17x
2
 + 17y
2
 - 68x - 102y + 221 = x
2
 + 16y
2
 + 6x - 24y - 8xy + 9
? 16x
2
 + y
2
 + 8xy - 74x - 78y + 212 = 0
?The equation of the parabola is 16x
2
 + y
2
 + 8xy - 74x - 78y + 212 = 0.
2. Question
Find the equation of the parabola whose focus is the point (2, 3) and directrix is the line x – 4y + 3 = 0. Also,
find the length of its latus - rectum.
Answer
Given that we need to find the equation of the parabola whose focus is S(2, 3) and directrix(M) is x - 4y + 3 =
0.
Page 5


25. Parabola
Exercise 25.1
1 A. Question
Find the equation of the parabola whose:
focus is (3, 0) and the directrix is 3x + 4y = 1
Answer
Given that we need to find the equation of the parabola whose focus is S(3, 0) and directrix(M) is 3x + 4y - 1
= 0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 25x
2
 + 25y
2
 - 150x + 225 = 9x
2
 + 16y
2
 - 6x - 8y + 24xy + 1
? 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
?The equation of the parabola is 16x
2
 + 9y
2
 - 24xy - 144x + 8y + 224 = 0
1 B. Question
Find the equation of the parabola whose:
focus is (1, 1) and the directrix is x + y + 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(1, 1) and directrix(M) is x + y + 1 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 2x
2
 + 2y
2
 - 4x - 4y + 4 = x
2
 + y
2
 + 2x + 2y + 2xy + 1
? x
2
 + y
2
 + 2xy - 6x - 6y + 3 = 0
?The equation of the parabola is x
2
 + y
2
 + 2xy - 6x - 6y + 3 = 0.
1 C. Question
Find the equation of the parabola whose:
focus is (0, 0) and the directrix is 2x - y – 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(0, 0) and directrix(M) is 2x - y - 1 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 5x
2
 + 5y
2
 = 4x
2
 + y
2
 - 4x + 2y - 4xy + 1
? x
2
 + 4y
2
 + 4xy + 4x - 2y - 1 = 0
?The equation of the parabola is x
2
 + 4y
2
 + 4xy + 4x - 2y - 1 = 0.
1 D. Question
Find the equation of the parabola whose:
focus is (2, 3) and the directrix is x - 4y + 1 = 0
Answer
Given that we need to find the equation of the parabola whose focus is S(2, 3) and directrix(M) is x - 4y + 3 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 17x
2
 + 17y
2
 - 68x - 102y + 221 = x
2
 + 16y
2
 + 6x - 24y - 8xy + 9
? 16x
2
 + y
2
 + 8xy - 74x - 78y + 212 = 0
?The equation of the parabola is 16x
2
 + y
2
 + 8xy - 74x - 78y + 212 = 0.
2. Question
Find the equation of the parabola whose focus is the point (2, 3) and directrix is the line x – 4y + 3 = 0. Also,
find the length of its latus - rectum.
Answer
Given that we need to find the equation of the parabola whose focus is S(2, 3) and directrix(M) is x - 4y + 3 =
0.
Let us assume P(x, y) be any point on the parabola.
We know that the point on the parabola is equidistant from focus and directrix.
We know that the distance between two points (x
1
, y
1
) and (x
2
, y
2
) is .
We know that the perpendicular distance from a point (x
1
, y
1
) to the line ax + by + c = 0 is .
? SP = PM
? SP
2
 = PM
2
? 
? 
? 
? 17x
2
 + 17y
2
 - 68x - 102y + 221 = x
2
 + 16y
2
 + 6x - 24y - 8xy + 9
? 16x
2
 + y
2
 + 8xy - 74x - 78y + 212 = 0
?The equation of the parabola is 16x
2
 + y
2
 + 8xy - 74x - 78y + 212 = 0.
We know that the length of the latus rectum is twice the perpendicular distance from the focus to the
directrix.
? 
? 
? 
?The length of the latus rectum is .
3 A. Question
Find the equation of the parabola, if
the focus is at (- 6, 6) and the vertex is at (- 2, 2)
Answer
We need to find the equation of the parabola whose focus is (- 6, 6), and the vertex is (- 2, 2).
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