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JEE • MATHEMATICS • PARABOLA

Parabola practice · Medium

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These sets use single-correct-answer MCQs. They do not cover every JEE Main or JEE Advanced question format and are not full-length mock papers.

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Showing 61-90 of 211 questions
61 Medium
61 of 211
Application +4 | -1
The tangent to the parabola y2=4axy^2=4 a x at the point (a,2a)(a, 2 a) makes with xx-axis an angle equal to
View answer and explanation
Correct option: B
π/4\pi / 4
Option B (π/4\pi / 4) is correct because the slope of the tangent line at (a, 2a) is 1, which corresponds to an angle of 45 degrees.
62 Medium
62 of 211
Analyzing +4 | -1
A variable tangent to the parabola y2=4ax{y^2} = 4ax meets the parabola y2=−4ax{y^2} = - 4ax at PP and QQ. The locus of the mid-point of PQPQ is
View answer and explanation
Correct option: C
y2−43ax{y^2} - \frac{4}{3}ax
Option C is correct because the locus of the midpoint is found by using the tangent equation and properties of the parabola.
63 Medium
63 of 211
Application +4 | -1
The point on the parabola y2=8xy^2=8 x at which the normal is parallel to the line x−2y+5=0x-2 y+5=0 is
View answer and explanation
Correct option: B
(1/2,−2)(1 / 2,-2)
Option B ((1/2,−2)(1 / 2,-2)) is correct because the point satisfies the parabola equation and the normal's slope condition.
64 Medium
64 of 211
Application +4 | -1
The normal at (a,2a)\left( {a,2a} \right) on y2=4ax,{y^2} = 4ax, meets the curve again at (at2,2at),\left( {a{t^2},2at} \right), then the value of tt is
View answer and explanation
Correct option: D
−3- 3
Option D (−3-3) is correct because it is found by using the formula relating the parameters of two points on a parabola where the normal at one point intersects the parabola at the other.
65 Medium
65 of 211
Application +4 | -1
If a normal of slope m to the parabola y2=4ax{y^2} = 4ax touches the hyperbola x2−y2=a2,{x^2} - {y^2} = {a^2}, then
View answer and explanation
Correct option: D
m6+4m4+3m2+1=0{m^6} + 4{m^4} + 3{m^2} + 1 = 0
Option B (900{90^0}) is correct because the point (5,4) lies on the director circle of the ellipse, implying that the tangents from that point are perpendicular.
66 Medium
66 of 211
Application +4 | -1
If the tangent at the point P(2,4)P(2,4) to the parabola y2=8xy^2=8 x meets the parabola y2=8x+5y^2=8 x+5 at QQ and RR, then the mid-point of QRQ R, is
View answer and explanation
Correct option: A
(2,4)(2,4)
Option A (2,4) is correct because the midpoint of the chord QR is the same as the point P where the tangent is drawn to the first parabola.
67 Medium
67 of 211
Application +4 | -1
If a focal chord of the parabola y2=axy^2=a x is 2x−y−8=02 x-y-8=0, then the equation of the directrix, is
View answer and explanation
Correct option: A
x+4=0x+4=0
Option A (x+4=0x+4=0) is correct because by finding the value of 'a' and using the directrix formula, we can determine the equation of the directrix.
68 Medium
68 of 211
Analyzing +4 | -1
The tangent at the point P(x1,y1)P\left(x_1, y_1\right) to the parabola y2=4axy^2=4 a x meets the parabola y2=4a(x+b)y^2=4 a(x+b) at QQ and RR, the coordinates of the mid-point of QRQ R are
View answer and explanation
Correct option: B
(x1,y1)\left(x_1, y_1\right)
Option B ((x1,y1)(x_1, y_1)) is correct because the midpoint of the chord QR on the second parabola has the same coordinates as the point P on the first parabola.
69 Medium
69 of 211
Application +4 | -1
The angle between the tangents drawn at the end points of the latus rectum of parabola y2=4axy^2=4 a x is
View answer and explanation
Correct option: D
π/2\pi / 2
Option D (π2\frac{\pi}{2}) is correct because the angle between the tangents drawn at the end points of the latus rectum of the parabola y2=4axy^2=4 a x is 90∘90^{\circ}.
70 Medium
70 of 211
Application +4 | -1
Which of the following points lie on the parabola x2=4ayx^2=4 a y ?
View answer and explanation
Correct option: D
x=2at,y=at2x=2 a t, y=a t^2
Option D (x=2at,y=at2x=2 a t, y=a t^2) is correct because substituting these parametric equations into the parabola's equation satisfies the equation.
71 Medium
71 of 211
Application +4 | -1
The locus of a point P(α,β)P(\alpha, \beta) moving under the condition that the line y=αx+βy=\alpha x+\beta is a tangent to the parabola y2=4axy^2=4 a x is
View answer and explanation
Correct option: B
xy=ax y=a
Option B (xy=axy=a) is correct because the condition for tangency to the parabola y2=4axy^2=4ax gives the relation αβ=a\alpha \beta = a, leading to the locus xy=axy=a.
72 Medium
72 of 211
Application +4 | -1
Length of the focal chord of the parabola, y2=4xy^2=4 x, which makes an angle of π4\frac{\pi}{4} with the axis of xx is:
View answer and explanation
Correct option: A
8
Option A (8) is correct because it is the length of the focal chord calculated using the given angle and parabola equation.
73 Medium
73 of 211
Application +4 | -1
The ends of the latusrectum of the conic x2+10x−16y+25=0{x^2} + 10x - 16y + 25 = 0 are
View answer and explanation
Correct option: C
(3,4),(−13,4)\left( {3,4} \right),\left( { - 13,4} \right)
The correct answer is C ((3,4),(-13,4)) because these are the coordinates of the endpoints of the latus rectum after transforming the given equation into standard parabolic form.
74 Medium
74 of 211
Application +4 | -1
Let PP be a point (1,0)(1,0) and QQ a point on the locus y2=8xy^2=8 x. The locus of mid-point of PQP Q is
View answer and explanation
Correct option: D
y2−4x+2=0y^2-4 x+2=0
Option D (y2−4x+2=0y^2-4 x+2=0) is correct because it represents the locus of the midpoint of PQ, derived by substituting the midpoint coordinates into the equation of the parabola.
75 Medium
75 of 211
Analyzing +4 | -1
If the tangents drawn from a point to the parabola y2=4axy^2=4 a x are normals to the parabola x2=4byx^2=4 b y, then
View answer and explanation
Correct option: C
a2≥8b2a^2 \geq 8 b^2
Option C (a2≥8b2a^2 \geq 8 b^2) is correct because the condition for the tangents to the parabola y2=4axy^2=4ax to be normals to the parabola x2=4byx^2=4by leads to this inequality based on the discriminant of a quadratic equation.

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76 Medium
76 of 211
Application +4 | -1
Consider curve y2=xy^2=x. At which point on the curve, tangent line makes an angle of 45∘45^{\circ} with xx-axis?
View answer and explanation
Correct option: A
(1/4,1/2)(1 / 4,1 / 2)
Option A ((1/4,1/2)(1 / 4,1 / 2)) is correct because the derivative of the curve y2=xy^2 = x gives the slope of the tangent, which is equal to tan⁡(45∘)=1\tan(45^{\circ}) = 1.
77 Medium
77 of 211
Application +4 | -1
The line xcos⁡α+ysin⁡α=px \cos \alpha+y \sin \alpha=p touches the parabola y2=y^2= 4a(x+a)4 a(x+a) if
View answer and explanation
Correct option: A
pcos⁡α+a=0p \cos \alpha+a=0
Option C (acos⁡α+p=0a \cos \alpha+p=0) is correct because it represents the condition for the line xcos⁡α+ysin⁡α=px \cos \alpha+y \sin \alpha=p to touch the parabola y2=4a(x+a)y^2=4 a(x+a).
78 Medium
78 of 211
Application +4 | -1
If any tangent to parabola y2=4xy^2=4 x cut the co-ordinate axes at AA and BB respectively, The locus of mid point of ABA B is y2=−xλy^2=-\frac{x}{\lambda}, then λ\lambda is
View answer and explanation
Correct option: A
2
Option A (2) is correct because finding the locus of the midpoint of the intercepts of a tangent to the parabola gives the correct value of lambda.
79 Medium
79 of 211
Application +4 | -1
If Q1 & Q2Q_1 \,\&\, Q_2 be the angle made by tangents to the axis of y2=4xy^2=4 x from point P &P \,\& if Q1+Q2=45∘Q_1+Q_2=45^{\circ} then locus of point P{P} is
View answer and explanation
Correct option: C
y=(x−1)y=(x-1)
Option C (y=(x-1)) is correct because using the tangent formula and the given condition Q1+Q2=45∘Q_1 + Q_2 = 45^{\circ}, we can derive the locus of point P as y = x-1.
80 Medium
80 of 211
Application +4 | -1
The locus of the mid-points of the chords of the parabola x2=4pyx^2=4py having slope mm, is a
View answer and explanation
Correct option: C
line parallel to yy-axis at a distance ∣2pm∣|2 \mathrm{pm}| from it
The correct answer is B because the locus of the midpoints of chords with slope m for the parabola x2=4pyx^2 = 4py is a horizontal line at y=x24py = \frac{{x^2}}{{4p}}.
81 Medium
81 of 211
Analyzing +4 | -1
If two parabolas y2=4axy^2=4 a x and x2=4byx^2=4 b y intersect at an angle π3\frac{\pi}{3} at a point other than the origin, then the value of (a/b)1/3+(b/a)1/3(a / b)^{1 / 3}+(b / a)^{1 / 3}, is
View answer and explanation
Correct option: C
32\frac{\sqrt{3}}{2}
Option C (32\frac{{\sqrt{3}}}{{2}}) is correct because it is the value obtained after calculating the angle of intersection between the two parabolas.
82 Medium
82 of 211
Application +4 | -1
If the normal to y2=12xy^2=12 x at the point (3,6)(3,6) meets the parabola once again at the point (27,−18)(27,-18), and the circle on the normal chord as diameter is
View answer and explanation
Correct option: D
x2+y2−30x+12y−27=0x^2+y^2-30 x+12 y-27=0
Option D (x2+y2−30x+12y−27=0x^2+y^2-30 x+12 y-27=0) is correct because it represents the equation of the circle with the normal chord as diameter, derived from the given points (3,6) and (27,-18).
83 Medium
83 of 211
Application +4 | -1
The tangent to y=ax2+bx+72y=a x^2+b x+\frac{7}{2} at M(1,2){M}(1,2) is parallel to the tangent at N(−2,2){N}(-2,2) on the curve y=x2+6x+10y=x^2+6 x+10 then (a2−b)\left(\frac{a}{2}-b\right) is equal to
View answer and explanation
Correct option: A
3
Option A (3) is correct because the slopes of the tangents at the given points are equal, which allows us to solve for the unknowns a and b, and then evaluate the expression (a/2 - b).
84 Medium
84 of 211
Application +4 | -1
If the normal at (1,2) on the parabola y2=4x{y^2} = 4x meets the parabola again at the point (t2,2t),\left( {{t^2},2t} \right), then the value of tt is
View answer and explanation
Correct option: C
−3- 3
Option C (−3- 3) is correct because it is calculated using the formula relating the parameters of the points where the normal to a parabola intersects the parabola again.
85 Medium
85 of 211
Application +4 | -1
Angle between the tangents to the curve y=x2−5x+6y=x^2-5 x+6 at the points (2,0)(2,0) and (3,0)(3,0) is
View answer and explanation
Correct option: B
π/2\pi / 2
Option B (π/2\pi / 2) is correct because the tangents at the given points are perpendicular, as their slopes are negative reciprocals of each other.
86 Medium
86 of 211
Analyzing +4 | -1
A normal is drawn to the parabola y2=4y^2=4 ax at a point PP other than the vertex. If it cuts the parabola at point QQ, then the least value of OQO Q, where OO is the vertex of the parabola, is
View answer and explanation
Correct option: A
46a4 \sqrt{6} a
Option A (46a4 \sqrt{6} a) is correct because the least value of OQ is found using AM >= GM inequality and the normal equation.
87 Medium
87 of 211
Application +4 | -1
If a focal chord of y2=16xy^2=16 x is a tangent to the circle (x−6)2+y2=2(x-6)^2+y^2=2, then the positive value of the slope of this chord is
View answer and explanation
Correct option: A
1
Option A (1) is correct because the slope of the focal chord which is tangent to the given circle is found to be 1.
88 Medium
88 of 211
Analyzing +4 | -1
Statement 1: y=mx−1my=m x-\frac{1}{m} is always a tangent to the parabola, y2=−4xy^{2}=-4 x for all non-zero values of mm.

Statement 2: Every tangent to the parabola, y2=−4xy^{2}=-4 x will meet its axis at a point whose abscissa is non-negative.
View answer and explanation
Correct option: D
Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1 .
Option D (Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1) is correct because both statements are true, but the second statement doesn't explain the first.
89 Medium
89 of 211
Application +4 | -1
An angle between the tangents drawn from point (1,4)(1,4) to the parabola y2=4xy^2=4 x is
View answer and explanation
Correct option: C
π3\frac{\pi}{3}
Option C (π3\frac{\pi}{3}) is correct because it is the angle between the tangents calculated using the slopes of the tangents from the given point to the parabola.
90 Medium
90 of 211
Application +4 | -1
If f(x+y)=f(x)⋅f(y)f(x+y)=f(x) \cdot f(y) for all x,y∈Rx, y \in R and f(1)=2f(1)=2 and ar=f(r)a_r=f(r), for r∈Nr \in N then the coordinates of a point on the parabola y2=8xy^2=8 x whose focal distance is 4 may be
View answer and explanation
Correct option: A
(a1,a2)\left(a_1, a_2\right)
Option A ((a1,a2)\left(a_1, a_2\right)) is correct because the point (2,4) lies on the parabola and corresponds to the values of a1a_1 and a2a_2 derived from the given function.
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