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JEE • MATHEMATICS • PARABOLA

Parabola practice · Mixed difficulty

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Showing 151-180 of 586 questions

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151 Easy
151 of 586
Application +4 | -1
The focus of the parabola y=2x2+xy = 2{x^2} + x is
View answer and explanation
Correct option: C
(−14,0)\left( { - \frac{1}{4},0} \right)
Option D ((-14\frac{1}{4},0)) is correct because it represents the focus of the given parabola after completing the square and transforming the equation.
152 Easy
152 of 586
Analyzing +4 | -1
The normals at the ends of the latusrectum of the parabola y2=4axy^2=4 a x intersect at angle of
View answer and explanation
Correct option: D
π2\frac{\pi}{2}
Option D (π2\frac{{\pi}}{{2}}) is correct because the normals at the ends of the latus rectum are perpendicular to each other.
153 Easy
153 of 586
Application +4 | -1
The slope of the normal at the point (at2,2at)\left( {a{t^2},2at} \right) of the parabola, y2=4ax,{y^2} = 4ax, is
View answer and explanation
Correct option: C
−t- t
Option C (-t) is correct because the slope of the normal is the negative reciprocal of the slope of the tangent, which is -t.
154 Easy
154 of 586
Application +4 | -1
If the normal at (ap2,2ap)\left( {a{p^2},2ap} \right) on the parabola y2=4ax{y^2} = 4ax, meets the parabola again at (aq2,2aq),\left( {a{q^2},2aq} \right), then
View answer and explanation
Correct option: A
p2+pq+2=0{p^2} + pq + 2 = 0
Option A (p2+pq+2=0{p^2} + pq + 2 = 0) is correct because it is derived from the condition for the normal at a point on the parabola to meet the parabola again.
155 Medium
155 of 586
Application +4 | -1
If end points t1,t2t_1, t_2 of a chord satisfy the relation t1t2=1t_1 t_2=1 then chord always passes through the point if parabola is y2=4x{y}^2=4 {x}
View answer and explanation
Correct option: B
(−1,0)(-1,0)
Option B ((-1,0)) is correct because the chord always passes through the point (-1,0) when the product of the parameters of the endpoints is 1.
156 Medium
156 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=4axy^2=4 a x is another parabola with directrix
View answer and explanation
Correct option: C
x=0x=0
Option C (x=0x=0) is correct because it represents the directrix of the new parabola formed by the midpoints, found by eliminating the parameter and identifying the directrix.
157 Medium
157 of 586
Application +4 | -1
The common tangent of the parabolas y2=4x{y^2} = 4x and x2=−8y{x^2} = - 8y is
View answer and explanation
Correct option: D
None of these
Option D (None of these) is correct because the common tangent equation derived does not match any of the given options.
158 Easy
158 of 586
Understanding +4 | -1
The maximum number of normal that can be drawn from a point to a parabola is
View answer and explanation
Correct option: D
3
Option D (3) is correct because a cubic equation can have at most three real roots, and the equation for the normal to a parabola is cubic.
159 Easy
159 of 586
Analyzing +4 | -1
If the tangents and normals at the extremities of a focal chord of a parabola intersect at (x1,y1)\left(x_1, y_1\right) and (x2,y2)\left(x_2, y_2\right) respectively, then
View answer and explanation
Correct option: C
y1=y2y_1=y_2
Option C (y1=y2y_1=y_2) is correct because the y-coordinates of the intersection points of tangents and normals at the extremities of a focal chord are equal.
160 Easy
160 of 586
Analyzing +4 | -1
If the axis of the parabola y2=4axy^2=4 a x, its directrix and the chord joining t1t_1 and t2t_2 are concurrent, then t1t2=t_1 t_2=
View answer and explanation
Correct option: B
1
Option A (1 / 2,1 / 2) is incorrect. The correct answer is C (-1) because it is derived from the condition that the axis, directrix, and chord joining the points are concurrent.
161 Easy
161 of 586
Analyzing +4 | -1
The curve described parametrically by x=t2+2t−1,  y=3t+5x = {t^2} + 2t - 1,\;y = 3t + 5 represents
View answer and explanation
Correct option: C
A parabola
Option C (A parabola) is correct because eliminating the parameter 't' results in an equation of a parabola.
162 Medium
162 of 586
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.
163 Medium
163 of 586
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.
164 Medium
164 of 586
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.
165 Medium
165 of 586
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.

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166 Medium
166 of 586
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.
167 Easy
167 of 586
Application +4 | -1
The latusrectum of a parabola whose focal chord is PSQP S Q such that SP=3S P=3 and SQ=2S Q=2, is given by
View answer and explanation
Correct option: A
24/524 / 5
Option A (24/524 / 5) is correct because the latus rectum is twice the semi-latus rectum, which is the harmonic mean of the segments of the focal chord.
168 Easy
168 of 586
Application +4 | -1
The equation of the latusrectum of the parabola x2+4x+2y=0{x^2} + 4x + 2y = 0, is equal to
View answer and explanation
Correct option: C
2y=32y = 3
Option C (2y=32y = 3) is correct because it represents the equation of the latus rectum after completing the square and shifting the parabola to its standard form.
169 Easy
169 of 586
Application +4 | -1
The vertex of the parabola x2+8x+12y+4=0x^2+8 x+12 y+4=0 is
View answer and explanation
Correct option: A
(−4,1)(-4,1)
Option A ((-4, 1)) is correct because the vertex of the parabola is (-4, 1) after rewriting the equation in standard form.
170 Easy
170 of 586
Application +4 | -1
On the parabola y=x2y = {x^2}, the point at least distance from the straight line y=2x−4y = 2x - 4 is
View answer and explanation
Correct option: A
(1,1)\left( {1,1} \right)
Option A ((1,1)) is correct because it represents the point on the parabola where the tangent is parallel to the given line, minimizing the distance.
171 Easy
171 of 586
Application +4 | -1
The tangents drawn from the ends of latusrectum of y2=12x{y^2} = 12x meets at
View answer and explanation
Correct option: A
Directrix
Option A (Directrix) is correct because the tangents at the ends of the latus rectum of a parabola intersect at the directrix.
172 Easy
172 of 586
Application +4 | -1
The intersection point of the normals drawn at the end points of latusrectum of the parabola x2=−2y{x^2} = - 2y is
View answer and explanation
Correct option: D
(0,−32)\left( {0, - \frac{3}{2}} \right)
Option D (0,−32)\left( {0, - \frac{{3}}{{2}}} \right) is correct because it is the intersection point obtained by solving the equations of the normals at the end points of the latus rectum.
173 Medium
173 of 586
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.
174 Medium
174 of 586
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.
175 Easy
175 of 586
Analyzing +4 | -1
Let the line lx+my=1lx+m y=1 cut the parabola y2=4axy^2=4 a x in the points AA and BB. Normals at AA and BB meet at point CC. Normal from CC other than these two meet the parabola at DD, then the coordinate of DD are
View answer and explanation
Correct option: D
(4am2l2,4aml)\left(\frac{4 a m^2}{l^2}, \frac{4 a m}{l}\right)
Option B (3−223-2 \sqrt{2}) is correct because it represents the minimum squared distance between the circle and the parabola, calculated using the common normal.
176 Easy
176 of 586
Application +4 | -1
If vertex of a parabola is (2,−1)(2,-1) and the equation of its directrix is 4x−3y=214 x-3 y=21, then the length of its latus rectum is:
View answer and explanation
Correct option: B
8
Option B (8) is correct because the length of the latus rectum is four times the distance from the vertex to the directrix.
177 Medium
177 of 586
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.
178 Easy
178 of 586
Application +4 | -1
A tangent to the parabola y2=8xy^2=8 x makes an angle of 45∘45^{\circ} with the straight line y=3x+5y=3 x+5, then the equation of tangent is
View answer and explanation
Correct option: C
2x+y+1=02 x+y+1=0
Option C (2x+y+1=02 x+y+1=0) is correct because it represents the equation of the tangent to the parabola y2=8xy^2=8 x that makes an angle of 45∘45^{\circ} with the straight line y=3x+5y=3 x+5.
179 Medium
179 of 586
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}.
180 Medium
180 of 586
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.
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