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JEE • MATHEMATICS • PARABOLA

Parabola practice · Easy

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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 301-330 of 356 questions

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301 Easy
301 of 356
Application +4 | -1
Equation of the directrix of the parabola whose focus is (0,0)(0,0) and the tangent at the vertex is x−y+1=0x-y+1=0 is
View answer and explanation
Correct option: C
x−y+2=0x-y+2=0
Option C (x−y+2=0x-y+2=0) is correct. The equation of the directrix is found by using the distance between the focus and the directrix, which is twice the distance between the focus and the tangent at the vertex.
302 Easy
302 of 356
Application +4 | -1
The focus of the parabola 4y2+12x−20y4 y^2+12 x-20 y +67=0+67=0 is
View answer and explanation
Correct option: C
(−17/4,5/2)(-17 / 4,5 / 2)
Option C ((-17 / 4,5 / 2)) is correct because it represents the focus of the given parabola, found by completing the square and transforming the equation to standard form.
303 Easy
303 of 356
Application +4 | -1
If x−2y−a=0x - 2y - a = 0 is a chord of y2=4ax,{y^2} = 4ax, then its length is
View answer and explanation
Correct option: C
20 a20\,a
Option C (20 a20\,a) is correct because the given line is a focal chord, and its length can be calculated using the formula 4acsc⁡2θ4a \csc^2 \theta.
304 Easy
304 of 356
Application +4 | -1
The latus rectum of the parabola y2−4y−2x−8=0y^2-4 y-2 x-8=0 is
View answer and explanation
Correct option: A
2
Option A (2) is correct because the latus rectum of the parabola is 2 after rewriting the equation in standard form.
305 Easy
305 of 356
Application +4 | -1
The normal to the parabola y2=4xy^2=4 x at P(1,2)P(1,2) meets the parabola again in QQ, then the coordinates of QQ are
View answer and explanation
Correct option: B
(9,−6)(9,-6)
Option B ((9,-6)) is correct because using the formula to find the other point of intersection of the normal with the parabola yields (9,-6).
306 Easy
306 of 356
Application +4 | -1
If the axis of a parabola is horizontal and it passes through the points (0,0),(0,−1)(0,0),(0,-1) and (6,1)(6,1), then its equation is
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the general form of a parabola with a horizontal axis is x=Ay2+By+Cx = Ay^2 + By + C, and none of the given options fit this form after substituting the points.
307 Easy
307 of 356
Analyzing +4 | -1
The parametric representation (2+t2,2  t+1)\left( {2 + {t^2},2\;t + 1} \right) represents
View answer and explanation
Correct option: B
A parabola with vertex at (2,1)\left( {2,1} \right)
Option D (Finitely may solutions) is correct because by solving the system of equations, we find that there are a finite number of solutions for x, y, and z.
308 Easy
308 of 356
Analyzing +4 | -1
If the parabolas y2=4b(x−c){y^2} = 4b(x - c) and y2=8ax {y^2} = 8ax\, have a common normal, then which one of the following is a valid choice for the ordered triad (a, b, c)?
View answer and explanation
Correct option: C
(1, 1, 3)
Option C (1, 1, 3) is the correct choice because it satisfies the condition for the parabolas to have a common normal other than the axis.
309 Easy
309 of 356
Analyzing +4 | -1
If parabolas y2=axy^2=a x and 25[(x−3)2+(y+2)2]=(3x−4y−2)225\left[(x-3)^2+(y+2)^2\right]=(3 x-4 y-2)^2 are equal, then the value of aa is
View answer and explanation
Correct option: B
6
Option B (6) is correct because the value of 'a' is determined by equating the latus rectum of both parabolas.
310 Easy
310 of 356
Application +4 | -1
The equations of the normals at the ends of latus rectum of the parabola y2=4axy^2=4 a x are given by
View answer and explanation
Correct option: A
x2−y2−6ax+9a2=0x^2-y^2-6 a x+9 a^2=0
Option D (y=m(x−a)−2am−am3y=m(x-a)-2 a m-a m^3) is correct because it's the correct form of the normal equation adjusted for the shifted parabola.
311 Easy
311 of 356
Application +4 | -1
Equation of chord of the parabola y2=16x{y^2} = 16x whose mid point is (1,1)\left( {1,1} \right), is
View answer and explanation
Correct option: D
8x−y=78x - y = 7
Option D (8x−y=78x - y = 7) is correct because it is the equation of the chord of the parabola with the given midpoint, found using T = S1.
312 Easy
312 of 356
Analyzing +4 | -1
If the normals to the parabola y2=4axy^2=4 a x at three points P,QP, Q and RR meet at AA and SS is the focus, then SP⋅SQ⋅SRS P \cdot S Q \cdot S R is equal to
View answer and explanation
Correct option: C
a(SA)2a(S A)^2
Option C (a(SA)2a(S A)^2) is correct because it relates the product of distances from the focus to the points where normals meet at a point A to the distance SA.
313 Easy
313 of 356
Analyzing +4 | -1
A normal drawn at a point PP on the parabola y2=4axy^2=4ax meets the curve again at QQ. The least distance of QQ from the axis of the parabola, is
View answer and explanation
Correct option: C
42a4 \sqrt{2} a
Option C (42a4 \sqrt{2} a) is correct because it represents the minimum distance of Q from the axis, calculated using AM >= GM inequality.
314 Easy
314 of 356
Application +4 | -1
The polar of the point (2,1)(2,1) with respect to the parabola y2=6xy^2=6 x, is
View answer and explanation
Correct option: B
y=3x+6y=3 x+6
Option B (y=3x+6y=3 x+6) is correct because it is the polar equation of the point (2,1) with respect to the parabola y2=6xy^2=6x.
315 Easy
315 of 356
Application +4 | -1
The parabola with directrix x+2y−1=0x + 2y - 1 = 0 and focus (1,0)\left( {1,0} \right) is
View answer and explanation
Correct option: A
4x2−4xy+y2−8x+4y+4=04{x^2} - 4xy + {y^2} - 8x + 4y + 4 = 0
Option A (4x2−4xy+y2−8x+4y+4=04{x^2} - 4xy + {y^2} - 8x + 4y + 4 = 0) is correct because it represents the equation of the parabola with the given directrix and focus.

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316 Easy
316 of 356
Analyzing +4 | -1
The equation y2−8y−x+19=0{y^2} - 8y-x + 19 = 0 represents a parabola whose focus is (14,0)\left( {\frac{1}{4},0} \right) and directrix is a parabola whose vertex is (3,4) and directrix is
View answer and explanation
Correct option: B
x=114x = \frac{{11}}{4}
Option B (x=114x = \frac{{11}}{4}) is correct because it is the directrix of the parabola, calculated by finding the directrix in the translated coordinate system and then converting back to the original coordinates.
317 Easy
317 of 356
Application +4 | -1
If the vertex of a parabola is (0,2)\left( {0,2} \right) and the extremities of latusrectum are (−6,4)\left( { - 6,4} \right) and (6,4),\left( {6,4} \right), then, its equation is
View answer and explanation
Correct option: C
x2−8y+16=0{x^2} - 8y + 16 = 0
The equation of the parabola is x2−8y+16=0{x^2} - 8y + 16 = 0 because it is derived from the vertex and latus rectum information.
318 Easy
318 of 356
Application +4 | -1
Equation of a common tangent to the parabola y2=4x{y^2} = 4x and the hyperbola xy=2xy = 2 is
View answer and explanation
Correct option: A
x+2y+4=0x + 2y + 4 = 0
Option A (x+2y+4=0x + 2y + 4 = 0) is correct because it represents the common tangent found by equating the tangent equations of the parabola and hyperbola.
319 Easy
319 of 356
Analyzing +4 | -1
The number of points on y2=4axy^2=4ax at which the normal chords drawn subtend a right angle at the vertex, is
View answer and explanation
Correct option: B
2
Option B (2) is correct because there are two points on the parabola where the normal chords drawn subtend a right angle at the vertex, derived from the condition t2 = 2.
320 Easy
320 of 356
Application +4 | -1
If the straight line lx+my+n=0lx + my + n = 0 touches the parabola, y2=4ax{y^2} = 4ax then
View answer and explanation
Correct option: B
nl=am2nl = a{m^2}
Option B (nl=am2nl = a{m^2}) is correct because it represents the condition for tangency of the line lx+my+n=0lx + my + n = 0 to the parabola y2=4ax{y^2} = 4ax.
321 Easy
321 of 356
Application +4 | -1
If P(at12,2at1)P\left( {at_1^2,2a{t_1}} \right) and Q(at22,2at2)Q\left( {at_2^2,2a{t_2}} \right) are two variable points on the curve y2=4  ax{y^2} = 4\;ax and PQPQ subtends a right angle at the vertex, then t1t2{t_1}{t_2} is equal to
View answer and explanation
Correct option: D
−4- 4
Option D (−4-4) is correct because the product of t1t_1 and t2t_2 is equal to −4-4 when PQPQ subtends a right angle at the vertex of the parabola.
322 Easy
322 of 356
Application +4 | -1
The locus of the point of intersection of perpendicular tangents to the parabola y2=4axy^2=4 a x, is
View answer and explanation
Correct option: B
x=−ax=-a
Option B (x=−ax=-a) is correct because the locus of the intersection of perpendicular tangents to a parabola is its directrix, which is x = -a for the parabola y2=4axy^2 = 4ax.
323 Easy
323 of 356
Application +4 | -1
The length of the latusrectum of the parabola x2−4x−8y+12=0{x^2} - 4x - 8y + 12 = 0 is
View answer and explanation
Correct option: C
8
Option C (8) is correct because it's the length of the latus rectum calculated after rewriting the parabola equation in standard form.
324 Easy
324 of 356
Application +4 | -1
The length of the common chord of the parabolas y2=x{y^2} = x and x2=y{x^2} = y and is
View answer and explanation
Correct option: C
2\sqrt 2
Option C 2\sqrt 2 is correct because it is the distance between the intersection points (0,0) and (1,1) of the two parabolas.
325 Easy
325 of 356
Application +4 | -1
Let the tangent to the parabola S:y2=2xS:{y^2} = 2x at the point P (2, 2) meet the X -axis at Q and normal at it meet the parabola S at the point R. Then, the area (in square units) of Δ  PQR\Delta \,\,\mathrm{PQR} is equal to
View answer and explanation
Correct option: A
252\frac{{25}}{2}
Option A (252\frac{{25}}{2}) is correct because it is the area of the triangle formed by the tangent, normal, and parabola at the specified points.
326 Easy
326 of 356
Application +4 | -1
The latusrectum of the parabola y2=4ax{y^2} = 4ax whose focal chord is PSQPSQ such that SP=3SP = 3 and SQ=2SQ = 2, is given by
View answer and explanation
Correct option: A
245\frac{{24}}{5}
Option A (245\frac{{24}}{{5}}) is correct because the latusrectum is twice the semi-latusrectum, which is the harmonic mean of the segments of the focal chord.
327 Easy
327 of 356
Application +4 | -1
A common tangent is drawn to the circle x2+y2=a2x^2+y^2=a^2 and the parabola y2=4bxy^2=4 b x. If the angle which this tangent makes with the axis of xx is π4\frac{\pi}{4}, then the relationship between aa and bb is (a,b>0)(a, b>0)
View answer and explanation
Correct option: A
b=2ab=\sqrt{2} a
Option B (b=2ab=\sqrt{2} a) is correct because it provides the relationship between a and b when the tangent to the parabola and circle is common and makes an angle of π4\frac{{\pi}}{{4}} with the x-axis.
328 Easy
328 of 356
Application +4 | -1
The point on the parabola y2=18xy^2=18 x, for which the ordinate is three times the abscissa, is
View answer and explanation
Correct option: D
(2,6)(2,6)
Option D ((2,6)(2,6)) is correct because substituting y = 3x into the parabola equation yields x = 2 and y = 6.
329 Easy
329 of 356
Application +4 | -1
The equation of the directrix of the parabola 25{(x−2)2+(y+5)2}=(3x+4y−1)225\left\{(x-2)^2+(y+5)^2\right\}=(3 x+4 y-1)^2, is
View answer and explanation
Correct option: B
3x+4y−1=03 x+4 y-1=0
Option B (3x+4y−1=03 x+4 y-1=0) is correct because the given equation is in the form SP = PM, where PM is the perpendicular distance from the point to the directrix.
330 Easy
330 of 356
Application +4 | -1
The vertex of the parabola whose parametric equation is x=t2−t+1,y=t2+t+1,t∈Rx=t^2-t+1, y=t^2+t+1, t \in R, is
View answer and explanation
Correct option: A
(1,1)(1,1)
Option A ((1,1)) is correct because it represents the vertex of the parabola defined by the given parametric equations.
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