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JEE • MATHEMATICS • PARABOLA

Parabola practice · Mixed difficulty

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Showing 91-120 of 586 questions

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91 Easy
91 of 586
Application +4 | -1
The equations x=t4,y=t24x=\frac{t}{4}, y=\frac{t^2}{4} represents
View answer and explanation
Correct option: B
A parabola
Option B (A parabola) is correct because eliminating the parameter 't' results in an equation of the form x2=kyx^2 = ky, which is a parabola.
92 Medium
92 of 586
Analyzing +4 | -1
The two parabolas y2=4x{y^2} = 4x and x2=4  y{x^2} = 4\;y intersect at a point PP, whose abscissae is not zero, such that
View answer and explanation
Correct option: C
The tangents to each curve at PP make complementary angles with the xx-axis
Option C is correct because the product of the slopes of the tangents at the point of intersection is 1, implying the angles are complementary.
93 Medium
93 of 586
Analyzing +4 | -1
If P(h,k)P(h, k) be a point on the parabola x=4y2x=4 y^2, which is nearest to the point Q(0,33)Q(0,33), then the distance of PP from the directrix of the parabola y2=4(x+y)y^2=4(x+y) is equal to
View answer and explanation
Correct option: B
6
The correct answer is B (6) because after finding the point P on the parabola closest to Q, its distance from the directrix of the transformed parabola is calculated.
94 Medium
94 of 586
Application +4 | -1
The focal distance of a point on the parabola y2=4xy^2=4 x, which lies above its axis, is 10 units. If coordinates of the point are (h,k)(h, k), then h+kh+k is equal to
View answer and explanation
Correct option: C
15
Option C (15) is correct because it is the sum of the coordinates of the point on the parabola, calculated using the focal distance formula.
95 Medium
95 of 586
Application +4 | -1
If x+y=kx+y=k is a normal to the parabola y2=12xy^2=12 x, then kk is
View answer and explanation
Correct option: B
9
Option B (9) is correct because it is the value of k for which the line x+y=k is a normal to the parabola y2=12x, derived from the normal equation.
96 Medium
96 of 586
Understanding +4 | -1
A set of the parallel chords of the parabola y2=4axy^2=4 a x have their mid-point on
View answer and explanation
Correct option: C
Any straight line parallel to the axis
Option C (Any straight line parallel to the axis) is correct because the locus of the midpoints of parallel chords of a parabola is a straight line parallel to the axis of the parabola.
97 Easy
97 of 586
Application +4 | -1
If ℓ\ell is the length of latus-rectum of the parabola 3x2−10x−5y−20=03 {x}^2-10 {x}-5 {y}-20=0, then ℓ\ell equals
View answer and explanation
Correct option: C
53\frac{5}{3}
Option C (53\frac{5}{3}) is correct. The length of the latus rectum is found by rewriting the equation in standard form and identifying the coefficient of the y-term.
98 Easy
98 of 586
Application +4 | -1
If (2,−8)(2,-8) is at an end of a focal chord of the parabola y2=32xy^2=32 x, then the other end of the chord, is
View answer and explanation
Correct option: A
(32,32)(32,32)
Option A ((32,32)) is correct because by using the property of focal chord t1t2=−1t_1t_2 = -1 and substituting the given point, we can find the other end of the chord.
99 Hard
99 of 586
Application +4 | -1
The chord of contact of tangents drawn from any point on x−1=0x - 1 = 0 to y2−6y+4x+9=0{y^2} - 6y + 4x + 9 = 0 passes through the point
View answer and explanation
Correct option: A
(−1,3)\left( { - 1,3} \right)
Option A ((-1,3)) is correct because the chord of contact of tangents drawn from any point on the directrix of a parabola passes through its focus.
100 Medium
100 of 586
Application +4 | -1
The point on the parabola y2=8xy^2=8 x at which the normal is inclined at 60∘60^{\circ} to xx-axis has the coordinates
View answer and explanation
Correct option: A
(6,−43)(6,-4 \sqrt{3})
Option A ((6,−43)(6,-4 \sqrt{3})) is correct because the point satisfies the parabola equation and the normal's inclination condition.
101 Easy
101 of 586
Application +4 | -1
The vertex of the parabola y2+2y+x=0y^2+2 y+x=0 lies in which quadrant?
View answer and explanation
Correct option: D
Fourth
Option D (Fourth) is correct because the vertex of the parabola (y+1)2=−(x−1)(y+1)^2=-(x-1) is (1, -1), which lies in the fourth quadrant.
102 Easy
102 of 586
Application +4 | -1
Focus and directrix of the parabola x2=−8ayx^2=-8 a y are
View answer and explanation
Correct option: A
(0,−2a)(0,-2 a) and y=2ay=2 a
Option A ((0,−2a)(0,-2 a) and y=2ay=2 a) is correct because these are the focus and directrix, respectively, for the given parabola.
103 Medium
103 of 586
Application +4 | -1
A chord is drawn through the focus of the parabola y2=6xy^{2}=6 x such that its distance from the vertex of this parabola is 52\frac{\sqrt{5}}{2}, then its slope can be:
View answer and explanation
Correct option: A
52\frac{\sqrt{5}}{2}
Option A (52\frac{\sqrt{5}}{2}) is correct because it represents a possible slope of the chord passing through the focus, derived from the given conditions.
104 Medium
104 of 586
Application +4 | -1
The area of the triangle formed by the tangents drawn from (x1,y1)\left(x_1, y_1\right) to y2=4axy^2=4ax and their chord of contact, is
View answer and explanation
Correct option: B
(y12−4ax1)3/22a\frac{\left(y_1{ }^2-4 a x_1\right)^{3 / 2}}{2 a}
Option B ((y12−4ax1)3/22a\frac{\left(y_1{ }^2-4 a x_1\right)^{3 / 2}}{2 a}) is correct because it represents the area of the triangle formed by the tangents and their chord of contact, derived using geometric properties and the equation of the parabola.
105 Medium
105 of 586
Application +4 | -1
The line y=2x+cy=2 x+c is a tangent to the parabola y2=16xy^2=16 x, if c equals
View answer and explanation
Correct option: D
2
Option D (2) is correct because when the line y=2x+cy=2x+c is tangent to the parabola y2=16xy^2=16x, the value of c is 2.

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106 Easy
106 of 586
Application +4 | -1
The equation of the directrix of the parabola x2−4x−3y+10=0,{x^2} - 4x - 3y + 10 = 0, is
View answer and explanation
Correct option: B
y=54y = \frac{5}{4}
Option B (y=54y = \frac{5}{4}) is correct because it represents the directrix of the given parabola after rewriting it in standard form.
107 Medium
107 of 586
Application +4 | -1
The tangents drawn at the ends of any focal chord of a parabola y2=4axy^2=4 a x intersect in the line
View answer and explanation
Correct option: D
x+a=0x+a=0
Option D ((18,−12)(18,-12)) is correct because it is the point where the normal at (2,4) intersects the parabola again.
108 Easy
108 of 586
Analyzing +4 | -1
Let A,B,CA, B, C be three points on the parabola y2=4axy^2=4 a x such that tangents at these points taken in pairs form a triangle PQRP Q R. Then, area (△ABC)(\triangle A B C) : area (△PQR)=(\triangle P Q R)=
View answer and explanation
Correct option: B
2:12: 1
Option B (2:1) is correct. The ratio of the area of the triangle formed by three points on a parabola to the area of the triangle formed by the tangents at those points is always 2:1.
109 Medium
109 of 586
Application +4 | -1
Number of normals drawn from the point (−2,2)(-2,2) to the parabola y2−2y−2x−1=0y^2-2 y-2 x-1=0 is
View answer and explanation
Correct option: A
One
Option A (One) is correct because the cubic equation for the slope of the normal has only one real root, indicating only one normal can be drawn.
110 Easy
110 of 586
Application +4 | -1
The point of intersection of the normals to the parabola y2=4xy^{2}=4 x at the ends of its latus rectum is :
View answer and explanation
Correct option: B
(3,0)(3,0)
Option B ((3,0)(3,0)) is correct because the point of intersection of the normals to the parabola y2=4axy^2 = 4ax at the ends of its latus rectum is (3a, 0).
111 Easy
111 of 586
Application +4 | -1
Vertex of the parabola is (2,−1)(2,-1) and equation of the directrix is 4x−3y=21.4 x-3 y=21. Then length of the latusrectum is
View answer and explanation
Correct option: B
8
Option B (8) is correct because the length of the latus rectum of a parabola is 4a, where a is the distance from the vertex to the directrix, and in this case, 4a = 8.
112 Medium
112 of 586
Application +4 | -1
If PP is a point on the parabola y2=4  ax{y^2} = 4\;ax such that the subtangent and subnormal at PP are equal, then the coordinates of PP are
View answer and explanation
Correct option: A
(a,2a)\left( {a,2a} \right) or (a,−2a)\left( {a, - 2a} \right)
Option A ((a,2a)\left( {a,2a} \right) or (a,−2a)\left( {a, - 2a} \right)) is correct because the subtangent equals the subnormal only at points where the ordinate is equal to the semi-latus rectum.
113 Medium
113 of 586
Application +4 | -1
The point of intersection of the latus rectum and the axis of the parabola y2+4x+2y−8=0y^2+4 x+2 y-8=0
View answer and explanation
Correct option: A
(5/4,−1)(5 / 4,-1)
Option A (5/4, -1) is correct because the point of intersection is the focus of the parabola, found by rewriting the equation in standard form.
114 Easy
114 of 586
Application +4 | -1
If a chord which is not a tangent, of the parabola y2=16xy^2=16 x has the equation 2x+y=p2 x+y=p, and mid-point (h,k)(h, k), then which of the following is (are) possible value (s) of p,hp, h and kk ?
View answer and explanation
Correct option: D
p=2,h=3,k=−4p=2, h=3, k=-4
Option D (p=2,h=3,k=−4p=2, h=3, k=-4) is correct because these values satisfy the equation derived from the chord's midpoint and the parabola's equation.
115 Easy
115 of 586
Application +4 | -1
If the vertex of the parabola y=x2−16x+ky = {x^2} - 16x + k lies on xx-axis, then the value of kk is
View answer and explanation
Correct option: C
64
Option C (64) is correct because when k=64, the equation becomes a perfect square, indicating the vertex lies on the x-axis.
116 Medium
116 of 586
Application +4 | -1
Let PQ be a double ordinate of the parabola, y2=−4xy^{2}=-4 x, where P\mathrm{P} lies in the second quadrant. If R\mathrm{R} divides PQ\mathrm{PQ} in the ratio 2:12: 1 then the locus of R\mathrm{R} is :
View answer and explanation
Correct option: D
9y2=−4x9 y^{2}=-4 x
Option D (9y2=−4x9 y^{2}=-4 x) is correct because it represents the locus of point R, derived from the given parabola and the section ratio.
117 Easy
117 of 586
Analyzing +4 | -1
Consider the two curves
C1:y2=4x,C2:x2+y2−6x+1=0{C_1}:{y^2} = 4x,{C_2}:{x^2} + {y^2} - 6x + 1 = 0
Then
View answer and explanation
Correct option: B
C1C_1 and C2C_2 touch each other exactly at two point
Option B is correct. The curves C1C_1 and C2C_2 touch each other exactly at two points, which are (1, 2) and (1, -2), as they have the same tangent at these points.
118 Medium
118 of 586
Analyzing +4 | -1
Let focus of a parabola be α,β\alpha, \beta such that the lines x+y−3=0,x−y+3=0,x+2y−3=0x+y-3=0, x-y+3=0, x+2 y-3=0 and x−2y+3=0x-2 y+3=0 are tangents to the parabola, then the value of α−β\alpha-\beta is equal to
View answer and explanation
Correct option: B
1
Option B (1) is correct because the focus of the parabola is found to be (0, -1) by solving the equations of the tangents, leading to α−β=0−(−1)=1\alpha - \beta = 0 - (-1) = 1.
119 Easy
119 of 586
Analyzing +4 | -1
The slope of a chord of the parabola y2=4axy^2=4 a x which is normal at one end and which subtends a right angle at the origin, is
View answer and explanation
Correct option: B
2\sqrt{2}
Option B (2\sqrt{2}) is correct because the slope of the normal chord is ±2\pm \sqrt{2} when it subtends a right angle at the origin.
120 Easy
120 of 586
Application +4 | -1
If tangents at extremities of a focal chord ABAB of the parabola y2=4ax{y^2} = 4ax intersect at a point CC, then ∠ACB\angle ACB is equal to
View answer and explanation
Correct option: C
π2\frac{\pi }{2}
Option C (π2\frac{\pi }{2}) is correct because the tangents at the extremities of a focal chord of a parabola intersect at right angles.
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