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JEE • MATHEMATICS • PARABOLA

Parabola practice · Mixed difficulty

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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 211-240 of 586 questions

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211 Medium
211 of 586
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).
212 Easy
212 of 586
Analyzing +4 | -1
The slope of the line touching both the parabolas y2=4xy^2=4 x and x2=−32yx^2=-32 y, is
View answer and explanation
Correct option: C
1/21 / 2
Option C (1/2) is correct because it is the slope that makes the tangent to the first parabola also tangent to the second.
213 Easy
213 of 586
Application +4 | -1
The point of contact of the line x−2y−1=0x - 2y - 1 = 0 with the parabola y2=2(x−3),{y^2} = 2\left( {x - 3} \right), is
View answer and explanation
Correct option: A
(5,2)\left( {5,2} \right)
Option A (5,2)\left( {5,2} \right) is correct because it represents the point of contact between the given line and the parabola, which can be found by comparing the given tangent equation with the standard form.
214 Medium
214 of 586
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).
215 Easy
215 of 586
Application +4 | -1
The line among the following which touches the parabola y2=4ax{y^2} = 4ax, is
View answer and explanation
Correct option: B
x−my+am2=0x - m\mathrm{y} + a{m^2} = 0
Option B (x−my+am2=0x - m\mathrm{y} + a{m^2} = 0) is correct because it represents the equation of a tangent to the parabola y2=4ax{y^2} = 4ax when the parameter is replaced with 1m\frac{1}{m}.
216 Easy
216 of 586
Application +4 | -1
If (9a,  6a)\left( {9a,\;6a} \right) is a point bounded in region formed by parabola y2=16x{y^2} = 16x and x=9x = 9, then
View answer and explanation
Correct option: A
a∈(0,1)a \in \left( {0,1} \right)
Option A (a∈(0,1)a \in \left( {0,1} \right)) is correct because it represents the range of 'a' that satisfies the condition for the point to be within the region bounded by the parabola and the line.
217 Easy
217 of 586
Application +4 | -1
Two mutually perpendicular tangent of the parabola y2=4axy^2=4 a x meet the axis at P1P_1 and P2P_2. It SS is the focus of the parabola, then 1SP1+1SP2\frac{1}{S P_1}+\frac{1}{S P_2} is equal to
View answer and explanation
Correct option: B
1a\frac{1}{a}
Option B (1a\frac{{1}}{{a}}) is correct because it represents the sum of the reciprocals of the distances from the focus to the points where mutually perpendicular tangents meet the axis.
218 Easy
218 of 586
Application +4 | -1
If two tangents drawn from a point PP to the parabola y2=4xy^2=4 x are at right angles, then the locus of PP, is
View answer and explanation
Correct option: A
x=−1x=-1
Option A (x=−1x=-1) is correct because the locus of the point of intersection of perpendicular tangents to a parabola is its directrix, and for y2=4xy^2=4x, the directrix is x=−1x=-1.
219 Medium
219 of 586
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.
220 Easy
220 of 586
Application +4 | -1
Three normals to the parabola y2=x{y^2} = x through point (a,0).\left( {a,0} \right). Then,
View answer and explanation
Correct option: C
a>12a > \frac{1}{2}
Option C (a>12a > \frac{1}{2}) is correct because for three normals to exist from a point (a, 0) to the parabola y2=x{y^2} = x, the condition a>12a > \frac{1}{2} must be satisfied.
221 Medium
221 of 586
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.
222 Easy
222 of 586
Application +4 | -1
The axis of the parabola x2−4x−3y+10=0x^2-4 x-3 y+10=0 is
View answer and explanation
Correct option: D
x−2=0x-2=0
Option D (x−2=0x-2=0) is correct because the axis of the parabola x2−4x−3y+10=0x^2-4x-3y+10=0 can be found by completing the square and identifying the axis of symmetry.
223 Medium
223 of 586
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.
224 Medium
224 of 586
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.
225 Medium
225 of 586
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.

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226 Medium
226 of 586
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.
227 Easy
227 of 586
Application +4 | -1
If two tangents drawn from a point PP to the parabola y2=4x{y^2} = 4x are at right angles, then the locus of PP is
View answer and explanation
Correct option: C
x=−1x = - 1
Option C (x=−1x = - 1) is correct because the locus of the point from which two perpendicular tangents are drawn to the parabola is the directrix of the parabola.
228 Medium
228 of 586
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.
229 Easy
229 of 586
Analyzing +4 | -1
If y=m1x+c1y = {m_1}x + {c_1} and y=m2x+c2,  m1≠m2y = {m_2}x + {c_2},\,\,{m_1} \ne {m_2} are two common tangents of circle x2+y2=2{x^2} + {y^2} = 2 and parabola y2=x,{y^2} = x, then the value of 8∣m1m2∣8\left| {{m_1}{m_2}} \right| is equal to
View answer and explanation
Correct option: C
−4+32- 4 + 3\sqrt 2
The correct answer is C (−4+32- 4 + 3\sqrt 2) because it is the value of 8∣m1m2∣8|m_1m_2| calculated using the condition for common tangents to the circle and parabola.
230 Medium
230 of 586
Application +4 | -1
The Cartesian equation of the directrix of the parabola whose parametric equations are x=2t+1,y=t2+2,x = 2t + 1,y = {t^2} + 2, is
View answer and explanation
Correct option: B
y=1y = 1
Option B (y = 1) is correct because by eliminating the parameter t, we get the parabola's equation, and then the directrix is found using the standard form.
231 Easy
231 of 586
Analyzing +4 | -1
Minimum distance between the parabolas y2−4x−8y+40=0y^2-4 x-8 y+40=0 and x2−8x−4y+40=0x^2-8 x-4 y+40=0, is
View answer and explanation
Correct option: D
2\sqrt{2}
Option D (2\sqrt{2}) is correct because the minimum distance between the parabolas is the distance between the points where the tangents are parallel to the line of symmetry y=xy=x.
232 Easy
232 of 586
Application +4 | -1
If the tangents at the points PP and QQ on a parabola meet in TT, then SP,STS P, S T and SQS Q are in
View answer and explanation
Correct option: B
G.P
Option B (G.P) is correct because SP, ST, and SQ form a geometric progression based on the properties of tangents to a parabola.
233 Medium
233 of 586
Application +4 | -1
The line lx+my+n=0l x+m y+n=0 touches the parabola y2=4axy^2=4 a x if
View answer and explanation
Correct option: C
ln⁡=am2\ln =a m^2
Option C (ln⁡=am2\ln =a m^2) is correct because it represents the condition for the line lx+my+n=0lx + my + n = 0 to be tangent to the parabola y2=4axy^2 = 4ax.
234 Hard
234 of 586
Analyzing +4 | -1
If the 4th term in the expansion of (px+1x)n,n∈N\left(p x+\frac{1}{x}\right)^n, n \in N is 52\frac{5}{2} and three normals to the parabola y2=xy^2=x are drawn through a point (q,0)(q, 0), then
View answer and explanation
Correct option: B
q>pq>p
Option B (q>pq>p) is correct because the condition for three normals to a parabola to be drawn from a point (q,0) implies q > 2a, and we find p = 1/2 and a = 1/4, thus q > p.
235 Easy
235 of 586
Application +4 | -1
The set of values of α\alpha in the interval [π/2,3π/2][\pi / 2,3 \pi / 2], for which the point (sin⁡α,cos⁡α)(\sin \alpha, \cos \alpha) does not lie outside the parabola 2y2+x−2=02 y^2+x-2=0, is
View answer and explanation
Correct option: C
[π/2,5π/6]∪[π,3π/2][\pi / 2,5 \pi / 6] \cup[\pi, 3 \pi / 2]
Option C ([π/2,5π/6]∪[π,3π/2][\pi / 2,5 \pi / 6] \cup[\pi, 3 \pi / 2]) is correct because it represents the intervals where the point lies inside or on the parabola.
236 Easy
236 of 586
Application +4 | -1
If three distinct normals are drawn from (2k,0)(2 k, 0) to the parabola y2=4xy^2=4 x, then
View answer and explanation
Correct option: B
k>1k>1
Option B (k>1k>1) is correct because for three distinct normals to be drawn, the condition k>1k>1 must be satisfied.
237 Easy
237 of 586
Application +4 | -1
If mm is the slope of a common tangent of the parabola y2=16xy^2=16 x and the circle x2+y2=8x^2+y^2=8, then m2m^2 is equal to
View answer and explanation
Correct option: A
1
Option A (1) is correct because it represents the square of the slope of the common tangent to the parabola and the circle, derived from the tangency conditions.
238 Medium
238 of 586
Application +4 | -1
The locus of the mid point of the line segment joining the focus to a moving point on the parabola y2=4ax{y^2} = 4ax is another parabola with the directrix
View answer and explanation
Correct option: C
x=0x = 0
Option C (x=0x = 0) is correct because the directrix of the new parabola formed by the midpoints is x=0x=0.
239 Medium
239 of 586
Application +4 | -1
If the focus of a parabola divides a focal chord of the parabola in segments of length 3 and 2 , the length of the latus rectum of the parabola is
View answer and explanation
Correct option: D
24/524 / 5
Option D (24/524 / 5) is correct. The length of the latus rectum is calculated using the given information about the focal chord segments.
240 Medium
240 of 586
Analyzing +4 | -1
The shortest distance between the parabolas y2=4x{y^2} = 4x and y2=2x−6{y^2} = 2x - 6 is
View answer and explanation
Correct option: B
5\sqrt 5
Option B (5\sqrt 5) is correct because the shortest distance between the parabolas occurs along the common normal, and the calculated distance is 5\sqrt 5.
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