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Q: 4. Can you explain how to find the axis of symmetry for a parabola?

Ans. The axis of symmetry for a parabola represented by the equation y = ax² + bx + c can be found using the formula x = -b/(2a). This line vertically divides the parabola into two mirror-image halves. For parabolas in the form of y = a(x - h)² + k, the axis of symmetry is the vertical line x = h.

Q: 5. What is the significance of the vertex in the context of a parabola?

Ans. The vertex of a parabola, given by the coordinates (h, k) in the vertex form y = a(x - h)² + k, is a crucial point as it represents the maximum or minimum point of the parabola. For parabolas that open upwards, the vertex is the minimum point, while for those that open downwards, it is the maximum point. The vertex also lies on the axis of symmetry, which aids in graphing the parabola accurately.

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 Page 1


JEE Main Previous Year Questions 
(2025): Parabola 
 
Q1: The axis of a parabola is the line ?? = ?? and its vertex and focus are in the first 
quadrant at distances v ?? and ?? v ?? units from the origin, respectively. If the point 
(?? , ?? ) lies on the parabola, then a possible value of ?? is :- 
A. 4 
B. 9 
C. 3 
D. 8 
Ans: B 
Solution: 
 
Directrix ?? + ?? = 0 
???? = ????
v(1 - 2)
2
+ (?? - 2)
2
=
(1 + ?? )
v 2
2?? 2
+ 8 - 8?? + 2 = ?? 2
+ 1 + 2?? ?? 2
- 10?? + 9 = 0
?? = 9
 
 
 
Page 2


JEE Main Previous Year Questions 
(2025): Parabola 
 
Q1: The axis of a parabola is the line ?? = ?? and its vertex and focus are in the first 
quadrant at distances v ?? and ?? v ?? units from the origin, respectively. If the point 
(?? , ?? ) lies on the parabola, then a possible value of ?? is :- 
A. 4 
B. 9 
C. 3 
D. 8 
Ans: B 
Solution: 
 
Directrix ?? + ?? = 0 
???? = ????
v(1 - 2)
2
+ (?? - 2)
2
=
(1 + ?? )
v 2
2?? 2
+ 8 - 8?? + 2 = ?? 2
+ 1 + 2?? ?? 2
- 10?? + 9 = 0
?? = 9
 
 
 
Q2: Let P be the parabola, whose focus is (-?? , ?? ) and directrix is ?? ?? + ?? + ?? = ?? . 
Then the sum of the ordinates of the points on P , whose abscissa is -2 , is 
A. 
3
2
 
B. 
5
2
 
C. 
1
4
 
D. 
3
4
 
Ans: A 
Solution: Equation of parabola 
(?? + 2)
2
+ (?? - 1)
2
= (
2?? + ?? + 2
v 5
)
2
 
 
 
5[(?? + 2)
2
+ (?? - 1)
2
] = (2?? + ?? + 2)
2
 Put ?? = -2,5(?? - 1)
2
= (?? - 2)
2
5(?? 2
- 2?? + 1) = ?? 2
- 4?? + 4
 ? 4?? 2
- 6?? + 1 = 0 ? ?? 1
+ ?? 2
=
3
2
 
 
Q3: Let the focal chord ???? of the parabola ?? ?? = ?? ?? make an angle of ????
°
 with the 
positive ?? -axis, where P lies in the first quadrant. If the circle, whose one diameter is 
PS, S being the focus of the parabola, touches the ?? -axis at the point (?? , ?? ), then ?? ?? ?? 
Page 3


JEE Main Previous Year Questions 
(2025): Parabola 
 
Q1: The axis of a parabola is the line ?? = ?? and its vertex and focus are in the first 
quadrant at distances v ?? and ?? v ?? units from the origin, respectively. If the point 
(?? , ?? ) lies on the parabola, then a possible value of ?? is :- 
A. 4 
B. 9 
C. 3 
D. 8 
Ans: B 
Solution: 
 
Directrix ?? + ?? = 0 
???? = ????
v(1 - 2)
2
+ (?? - 2)
2
=
(1 + ?? )
v 2
2?? 2
+ 8 - 8?? + 2 = ?? 2
+ 1 + 2?? ?? 2
- 10?? + 9 = 0
?? = 9
 
 
 
Q2: Let P be the parabola, whose focus is (-?? , ?? ) and directrix is ?? ?? + ?? + ?? = ?? . 
Then the sum of the ordinates of the points on P , whose abscissa is -2 , is 
A. 
3
2
 
B. 
5
2
 
C. 
1
4
 
D. 
3
4
 
Ans: A 
Solution: Equation of parabola 
(?? + 2)
2
+ (?? - 1)
2
= (
2?? + ?? + 2
v 5
)
2
 
 
 
5[(?? + 2)
2
+ (?? - 1)
2
] = (2?? + ?? + 2)
2
 Put ?? = -2,5(?? - 1)
2
= (?? - 2)
2
5(?? 2
- 2?? + 1) = ?? 2
- 4?? + 4
 ? 4?? 2
- 6?? + 1 = 0 ? ?? 1
+ ?? 2
=
3
2
 
 
Q3: Let the focal chord ???? of the parabola ?? ?? = ?? ?? make an angle of ????
°
 with the 
positive ?? -axis, where P lies in the first quadrant. If the circle, whose one diameter is 
PS, S being the focus of the parabola, touches the ?? -axis at the point (?? , ?? ), then ?? ?? ?? 
is equal to : 
A. 15 
B. 25 
C. 30 
D. 20 
Ans: A 
Solution: 
 
tan 60
°
=
2t - 0
t
2
- 1
= v 3 ? t = v 3
 ? P(3,2v 3)
 
Circle : 
 (?? - 1)(?? - 3) + (?? - 0)(?? - 2v 3) = 0
 at ?? = 0
 ? 3 + ?? 2
- 2v 3?? = 0
 ? ?? = v 3 = ?? 5?? 2
= 15
 
 
Q4: Let the point ?? of the focal chord ???? of the parabola ?? ?? = ???? ?? be (?? , -?? ). If the 
focus of the parabola divides the chord PQ in the ratio ?? : ?? , ?????? (?? , ?? ) = ?? , then 
?? ?? + ?? ?? is equal to : 
A. 17 
B. 10 
Page 4


JEE Main Previous Year Questions 
(2025): Parabola 
 
Q1: The axis of a parabola is the line ?? = ?? and its vertex and focus are in the first 
quadrant at distances v ?? and ?? v ?? units from the origin, respectively. If the point 
(?? , ?? ) lies on the parabola, then a possible value of ?? is :- 
A. 4 
B. 9 
C. 3 
D. 8 
Ans: B 
Solution: 
 
Directrix ?? + ?? = 0 
???? = ????
v(1 - 2)
2
+ (?? - 2)
2
=
(1 + ?? )
v 2
2?? 2
+ 8 - 8?? + 2 = ?? 2
+ 1 + 2?? ?? 2
- 10?? + 9 = 0
?? = 9
 
 
 
Q2: Let P be the parabola, whose focus is (-?? , ?? ) and directrix is ?? ?? + ?? + ?? = ?? . 
Then the sum of the ordinates of the points on P , whose abscissa is -2 , is 
A. 
3
2
 
B. 
5
2
 
C. 
1
4
 
D. 
3
4
 
Ans: A 
Solution: Equation of parabola 
(?? + 2)
2
+ (?? - 1)
2
= (
2?? + ?? + 2
v 5
)
2
 
 
 
5[(?? + 2)
2
+ (?? - 1)
2
] = (2?? + ?? + 2)
2
 Put ?? = -2,5(?? - 1)
2
= (?? - 2)
2
5(?? 2
- 2?? + 1) = ?? 2
- 4?? + 4
 ? 4?? 2
- 6?? + 1 = 0 ? ?? 1
+ ?? 2
=
3
2
 
 
Q3: Let the focal chord ???? of the parabola ?? ?? = ?? ?? make an angle of ????
°
 with the 
positive ?? -axis, where P lies in the first quadrant. If the circle, whose one diameter is 
PS, S being the focus of the parabola, touches the ?? -axis at the point (?? , ?? ), then ?? ?? ?? 
is equal to : 
A. 15 
B. 25 
C. 30 
D. 20 
Ans: A 
Solution: 
 
tan 60
°
=
2t - 0
t
2
- 1
= v 3 ? t = v 3
 ? P(3,2v 3)
 
Circle : 
 (?? - 1)(?? - 3) + (?? - 0)(?? - 2v 3) = 0
 at ?? = 0
 ? 3 + ?? 2
- 2v 3?? = 0
 ? ?? = v 3 = ?? 5?? 2
= 15
 
 
Q4: Let the point ?? of the focal chord ???? of the parabola ?? ?? = ???? ?? be (?? , -?? ). If the 
focus of the parabola divides the chord PQ in the ratio ?? : ?? , ?????? (?? , ?? ) = ?? , then 
?? ?? + ?? ?? is equal to : 
A. 17 
B. 10 
C. 37 
D. 26 
Ans: A 
Solution: y
2
 = 16x; a = 4 focus S =(4,0) 
 
2?? 1
= -4
 ? 2(4)?? 1
= -4
 ? ?? 1
= -
1
2
 ? ?? 1
?? 2
= -1
 ? ?? 2
= 2
 ? ?? (?? ?? 2
2
, 2?? ?? 2
) = (16,16)
 
Let, S divides PQ internally in ?? : 1 ratio 
 ?
16?? - 4
?? + 1
= 0
?? =
1
4
=
?? ?? ? ?? 2
+ ?? 2
= 1 + 16 = 17
 
 
Q5: The radius of the smallest circle which touches the parabolas ?? = ?? ?? + ?? and ?? =
?? ?? + ?? is 
A. 
7v 2
2
 
B. 
7v 2
16
 
Page 5


JEE Main Previous Year Questions 
(2025): Parabola 
 
Q1: The axis of a parabola is the line ?? = ?? and its vertex and focus are in the first 
quadrant at distances v ?? and ?? v ?? units from the origin, respectively. If the point 
(?? , ?? ) lies on the parabola, then a possible value of ?? is :- 
A. 4 
B. 9 
C. 3 
D. 8 
Ans: B 
Solution: 
 
Directrix ?? + ?? = 0 
???? = ????
v(1 - 2)
2
+ (?? - 2)
2
=
(1 + ?? )
v 2
2?? 2
+ 8 - 8?? + 2 = ?? 2
+ 1 + 2?? ?? 2
- 10?? + 9 = 0
?? = 9
 
 
 
Q2: Let P be the parabola, whose focus is (-?? , ?? ) and directrix is ?? ?? + ?? + ?? = ?? . 
Then the sum of the ordinates of the points on P , whose abscissa is -2 , is 
A. 
3
2
 
B. 
5
2
 
C. 
1
4
 
D. 
3
4
 
Ans: A 
Solution: Equation of parabola 
(?? + 2)
2
+ (?? - 1)
2
= (
2?? + ?? + 2
v 5
)
2
 
 
 
5[(?? + 2)
2
+ (?? - 1)
2
] = (2?? + ?? + 2)
2
 Put ?? = -2,5(?? - 1)
2
= (?? - 2)
2
5(?? 2
- 2?? + 1) = ?? 2
- 4?? + 4
 ? 4?? 2
- 6?? + 1 = 0 ? ?? 1
+ ?? 2
=
3
2
 
 
Q3: Let the focal chord ???? of the parabola ?? ?? = ?? ?? make an angle of ????
°
 with the 
positive ?? -axis, where P lies in the first quadrant. If the circle, whose one diameter is 
PS, S being the focus of the parabola, touches the ?? -axis at the point (?? , ?? ), then ?? ?? ?? 
is equal to : 
A. 15 
B. 25 
C. 30 
D. 20 
Ans: A 
Solution: 
 
tan 60
°
=
2t - 0
t
2
- 1
= v 3 ? t = v 3
 ? P(3,2v 3)
 
Circle : 
 (?? - 1)(?? - 3) + (?? - 0)(?? - 2v 3) = 0
 at ?? = 0
 ? 3 + ?? 2
- 2v 3?? = 0
 ? ?? = v 3 = ?? 5?? 2
= 15
 
 
Q4: Let the point ?? of the focal chord ???? of the parabola ?? ?? = ???? ?? be (?? , -?? ). If the 
focus of the parabola divides the chord PQ in the ratio ?? : ?? , ?????? (?? , ?? ) = ?? , then 
?? ?? + ?? ?? is equal to : 
A. 17 
B. 10 
C. 37 
D. 26 
Ans: A 
Solution: y
2
 = 16x; a = 4 focus S =(4,0) 
 
2?? 1
= -4
 ? 2(4)?? 1
= -4
 ? ?? 1
= -
1
2
 ? ?? 1
?? 2
= -1
 ? ?? 2
= 2
 ? ?? (?? ?? 2
2
, 2?? ?? 2
) = (16,16)
 
Let, S divides PQ internally in ?? : 1 ratio 
 ?
16?? - 4
?? + 1
= 0
?? =
1
4
=
?? ?? ? ?? 2
+ ?? 2
= 1 + 16 = 17
 
 
Q5: The radius of the smallest circle which touches the parabolas ?? = ?? ?? + ?? and ?? =
?? ?? + ?? is 
A. 
7v 2
2
 
B. 
7v 2
16
 
C. 
7v 2
4
 
D. 
7v 2
8
 
Ans: D 
Solution: The given parabolas are symmetric about the line y = x 
 
Tangents at A&B must be parallel to ?? = ?? line, so slope of the tangents = 1 
(
????
????
)
min?? = 1 = (
????
????
)
min?? For point B, y = x
2
+ 2
 
????
????
 = 2?? = 1
?? =
1
2
? ?? =
9
4
 ? Point ?? = (
1
2
,
9
4
) ? Point A = (
9
4
,
1
2
)
 
AB =
v
(
1
2
-
9
4
)
2
+ (
9
4
-
1
2
)
2
 =
v
98
16
=
7v 2
4
 Radius ==
7v 2
8

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FAQs on JEE Main Previous Year Questions (2025): Parabola - Mathematics (Maths) for JEE Main & Advanced

1. What is the standard form of a parabola, and how can it be identified?

Ans. The standard form of a parabola is given by the equation y = ax² + bx + c, where 'a', 'b', and 'c' are constants. A parabola opens upwards if 'a' is positive and downwards if 'a' is negative. The vertex of the parabola can be found using the formula x = -b/(2a), which gives the x-coordinate of the vertex. The corresponding y-coordinate can be calculated by substituting this x value back into the equation.

2. How do you determine the focus and directrix of a parabola?

Ans. For a parabola in the form y = a(x - h)² + k, the focus is located at (h, k + 1/(4a)) and the directrix is the line y = k - 1/(4a). The focus represents a point from which distances to any point on the parabola are measured, while the directrix is a line perpendicular to the axis of symmetry, helping to define the parabola's shape.

3. What role does the discriminant play in identifying the nature of roots for the quadratic equation related to parabolas?

Ans. The discriminant of a quadratic equation ax² + bx + c = 0 is given by D = b² - 4ac. It helps in determining the nature of the roots: if D > 0, there are two distinct real roots; if D = 0, there is exactly one real root (the parabola touches the x-axis); and if D < 0, there are no real roots (the parabola does not intersect the x-axis).

4. Can you explain how to find the axis of symmetry for a parabola?

Ans. The axis of symmetry for a parabola represented by the equation y = ax² + bx + c can be found using the formula x = -b/(2a). This line vertically divides the parabola into two mirror-image halves. For parabolas in the form of y = a(x - h)² + k, the axis of symmetry is the vertical line x = h.

5. What is the significance of the vertex in the context of a parabola?

Ans. The vertex of a parabola, given by the coordinates (h, k) in the vertex form y = a(x - h)² + k, is a crucial point as it represents the maximum or minimum point of the parabola. For parabolas that open upwards, the vertex is the minimum point, while for those that open downwards, it is the maximum point. The vertex also lies on the axis of symmetry, which aids in graphing the parabola accurately.

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