# Parabolas Quiz

If (h, k) is a point on the axis of the parabola 2(x -1)2 + 2(y – 1)2 = (x + y + 2)2 from where three distinct normals may be drawn, then
(a) h > 2
(b) h < 4
(c) h > 8
(d) h < 8

Ans. (a)

If (2, 0) is the vertex & y – axis the directrix of a parabola, then its focus is:
(a) (2, 0)
(b) (- 2, 0)
(c) (4, 0)
(d) (- 4, 0)

Ans. (c)

The polar focus of parabola
(a) x-axis
(b) y-axis
(c) Directrix
(d) Latus rectum

Ans. (c)

The length of the chord of the parabola, y2 = 12x passing through the vertex & making an angle of 60o with the axis of x is:
(a) 8
(b) 4
(c) 16/3

Ans. (a)

The point of intersection of the latus rectum and axis of the parabola y2 + 4x + 2y – 8 = 0
(a) (5/4, –1)
(b) (9/4, –1)
(c) (7/2, 5/2)

Ans. (a)

Related: Volume and Surface area quiz

The length of the side of an equilateral triangle inscribed in the parabola, y2 = 4x so that one of its angular point is at the vertex is:
(a) 8
(b) 6
(c) 4
(d) 2

Ans. (a)

The focal distance of a point on the parabola y2 = 16x whose ordinate is twice the abscissa, is
(a) 6
(b) 8
(c) 10
(d) 12

Ans. (b)

The equation of the tangent to the parabola y = (x – 3)2 parallel to the chord joining the points
(3, 0) and (4, 1) is:
(a) 2 x – 2 y + 6 = 0
(b) 2 y – 2 x + 6 = 0
(c) 4 y – 4 x + 11 = 0
(d) 4 x – 4 y = 11

Ans. (d)

The locus of the point of intersection of the perpendicular tangents to the parabola x2 = 4ay is
(a) Axis of the parabola
(b) Directrix of the parabola
(c) Focal chord of the parabola
(d) Tangent at vertex to the parabola

Ans. (b)

An equation of a tangent common to the parabolas y2 = 4x and x2 = 4y is
(a) x – y + 1 = 0
(b) x + y – 1 = 0
(c) x + y + 1 = 0
(d) y = 0

Ans. (c)

The ends of the latus rectum of the conic x2 + 10x – 16y + 25 = 0 are
(a) (3, –4), (13, 4)
(b) (–3, –4), (13, –4)
(c) (3, 4), (–13, 4)
(d) (5, –8), (–5, 8)

Ans. (c)

AP & BP are tangents to the parabola, y2 = 4x at A & B. If the chord AB passes through a fixed point (- 1, 1) then the equation of locus of P is
(a) y = 2 (x – 1)
(b) y = 2 (x + 1)
(c) y = 2 x
(d) y2 = 2 (x – 1)

Ans. (a)

The H.M. of the segments of a focal chord of the parabola y2 = 4ax is
(a) 4a
(b) 2a
(c) a
(d) a2

Ans. (b)

The equation of the normal to the parabola, y2 = 4ax at its point (am2, 2 am) is:
(a) y = – mx + 2am + am3
(b) y = mx – 2am – am3
(c) y = mx + 2am + am3
(d) none

Ans. (a)

Related: questions on Lines and Angles

The two parabolas y2 = 4x  and x2 = 4y intersect at a point P, whose abscissa is not zero, such that
(a) They both touch each other at P
(b) They cut at right angles at P
(c) The tangents to each curve at P make complementary angles with the x-axis

Ans. (c)

At what point on the parabola y2 = 4x the normal makes equal angles with the axes?
(a) (4, 4)
(b) (9, 6)
(c) (4, – 1)
(d) (1,  2)

Ans. (d)

A parabola passing through the point (–4, –2) has its vertex at the origin and y-axis as its axis. The latus rectum of the parabola is
(a) 6
(b) 8
(c) 10
(d) 12

Ans. (b)

If on a given base, a triangle be described such that the sum of the tangents of the base angles is a constant, then the locus of the vertex is:
(a) a circle
(b) a parabola
(c) an ellipse
(d) a hyperbola

Ans. (b)

If the line y = 2x + k is a tangent to the curve x2 = 4y, then k is equal to
(a) 4
(b) ½
(c) –4
(d) –1/2

Ans. (c)

Related: Discrete math quiz

A point moves such that the square of its distance from a straight line is equal to the difference between the square of its distance from the center of a circle and the square of the radius of the circle. The locus of the point is:
(a) a straight line at right angles to the given line
(b) a circle concentric with the given circle
(c) a parabola with its axis parallel to the given line
(d) a parabola with its axis perpendicular to the given line.

Ans. (d)

Two perpendicular tangents to y2 = ax always intersect on the line, if
(a) x = a
(b) x + a = 0
(c) x + 2a = 0
(d) x + 4a = 0

Ans. (b)

If the distances of two points P & Q from the focus of a parabola y2 = 4ax are 4 & 9, then the distance of the point of intersection of tangents at P & Q from the focus is:
(a) 8
(b) 6
(c) 5
(d) 13

Ans. (b)

PQ is any focal chord of the parabola y2 = 32x. The length of PQ can never be less than
(a) 8 units
(b) 16 units
(c) 32 units
(d) 48 units

Ans. (c)

From the point (4, 6) a pair of tangent lines are drawn to the parabola, y2 = 8x. The area of the triangle formed by these pair of tangent lines & the chord of contact of the point (4, 6) is:
(a) 8
(b) 4
(c) 2

Ans. (c)

Related: Euclid’s geometry questions

The tangent drawn at any point P to the parabola y2 = 4ax meets the directrix at point K, then the angle which KP subtends at its focus is
(a) 30o
(b) 45o
(c) 60o
(d) 90o

Ans. (d)

Locus of the intersection of the tangents at the ends of the normal chords of the parabola
y2 = 4ax is
(a) (2a + x)  y2 + 4a3 = 0
(b) (2a + x) + y2 = 0
(c) (2a + x)  y2 + 4a = 0
(d) none of these

Ans. (a)

The length intercepted by the curve y2 = 4x on the line satisfying dy/dx = 1 and passing through point (0, 1) is given by
(a) 1
(b) 2
(c) 0

Ans. (c)

Tangents are drawn from the points on the line x – y + 3 = 0 to parabola y2 = 8x. Then all the chords of contact passes through a fixed point whose coordinates are:
(a) (3, 2)
(b) (2, 4)
(c) (3, 4)
(d) (4, 1)

Ans. (c)

The points of intersection of the curves whose parametric equations are x = t2 + 1, y = 2t and x = 2s, y = 2/s is given by
(a) (1, –3)
(b) (2, 2)
(c) (–2, 4)
(d) (1, 2)

Ans. (b)

Related: Mensuration aptitude questions

The distance between a tangent to the parabola y2 = 4 A x (A > 0) and the parallel normal with gradient 1 is:
(a) 4 A
(b) 2 A
(c) 2 A
(d) A

Ans. (b)

The angle between two curves y2 = 4(x + 1) and x2 = 4(y + 1) is
(a) 0o
(b) 90o
(c) 60o
(d) 30o

Ans. (b)

AB is a chord of the parabola y2 = 4ax with vertex at A. BC is drawn perpendicular to AB meeting the axis at C. The projection of BC on the axis of the parabola is
(a) a
(b) 2a
(c) 4a
(d) 8a

Ans. (c)

The length of the latus rectum of the parabola x = ay2 + by + c is
(a) a/4
(b) a/3
(c) 1/a
(d) 1/4a

Ans. (c)

T is a point on the tangent to a parabola y2 = 4ax at its point P. TL and TN are the perpendiculars on the focal radius SP and the directrix of the parabola respectively. Then:
(a) SL = 2 (TN)
(b) 3 (SL) = 2 (TN)
(c) SL = TN
(d) 2 (SL) = 3 (TN)

Ans. (c)

Related: Probability Quiz

Tangents at the extremities of any focal chord of a parabola intersect
(a) At right angles
(b) On the directrix
(c) On the tangents at vertex

Ans. (a, b)

If from a variable point ‘P’ pair of perpendicular tangents PA and PB are drawn to any parabola then
(a) P lies on directrix of parabola
(b) chord of contact AB passes through focus
(c) chord of contact AB passes through of fixed point
(d) P lies on director circle

Ans. (a, b, c, d)

The equation of the other normal to the parabola y2 = 4ax which passes through the intersection of those at (4a, – 4a) & (9a, – 6a) is:
(a) 5x – y + 115 a = 0
(b) 5x + y – 135 a = 0
(c) 5x – y – 115 a = 0
(d) 5x + y + 115 = 0

Ans. (b)

The equation of the locus of a point which moves so as to be at equal distances from the point (a, 0) and the y-axis is
(a) y2 – 2ax + a2 = 0
(b) y2 + 2ax + a2 = 0
(c) x2 – 2ay + a2 = 0
(d) x2 + 2ay + a2 = 0

Ans. (a)

Related: Percentages problems

If x + y = k, is the normal to y2 = 12x, then k is
(a) 3
(b) 9
(c) –9
(d) – 3

Ans. (b)

A set of parallel chords of the parabola y2 = 4ax have their mid-point on
(a) Any straight line through the vertex
(b) Any straight line through the focus
(c) Any straight line parallel to the axis
(d) Another parabola

Ans. (c)

The focal chord to y2 = 16 x is tangent to (x – 6)2 + y2 = 2, then the possible values of the slope of this chord are:
(a) {- 1, 1}
(b) {- 2, 2}
(c) {- 2, 1/2}
(d) {2, – 1/2}

Ans. (a)

The order of the differential equation of all parabolas having directrix parallel to x-axis is
(a) 3
(b) 1
(c) 4
(d) 2

Ans. (a)

Related: mcq on Compound Interest

Let P be the point (1, 0) and Q a point of the locus y2 =  8x. The locus of mid point of PQ is
(a) x2 + 4y + 2 = 0
(b) x2 – 4y + 2 = 0
(c) y2 – 4x + 2 = 0
(d) y2 + 4x + 2 = 0

Ans. (c)

Two parabolas have the same focus. If their directrices are the x – axis & the y – axis respectively, then the slope of their common chord is:
(a) 1
(b) – 1
(c) 4/3
(d) ¾

Ans. (a, b)

If the normals at two points P and Q of a parabola y2 = 4ax intersect at a third point R on the curve, then the product of ordinates of P and Q is
(a) 4a2
(b) 2a2
(c) –4a2
(d) 8a2

Ans. (d)

A circle touch the line L and the circle C1 externally such that both the circles are on the same side of the line, then the locus of centre of the circle is
(a) ellipse
(b) hyperbola
(c) parabola
(d) parts of straight line

Ans. (c)

The parametric equation of the curve y2 = 8x are
(a) x = t2, y = 2t
(b) x = 2t2, y = 4t
(c) x = 2t, y = 4t2

Ans. (b)

Related: Ratios and proportions problems

If one end of a focal chord of the parabola y2 = 4x is (1, 2), the other end lies on
(a) x2 y + 2 = 0
(b) xy + 2 = 0
(c) xy – 2 = 0
(d) x2 + xy – y – 1 = 0

Ans. (a, b, d)

The centroid of the triangle formed by joining the feet of the normals drawn from any point to the parabola y2 = 4ax, lies on
(a) Axis
(b) Directrix
(c) Latusrectum
(d) Tangent at vertex

Ans. (a)

P is a point on the parabola y2 = 4ax (a > 0) whose vertex is A. PA is produced to meet the directrix in D and M is the foot of the perpendicular from P on the directrix. If a circle is described on MD as a diameter then it intersects the x-axis at a point whose co-ordinates are:
(a) (- 3a, 0)
(b) (- a, 0)
(c) (- 2a, 0)
(d) (a, 0)

Ans. (a, d)

The tangents at the extremities of a focal chord of a parabola
(a) are perpendicular
(b) are parallel
(c) intersect on the directrix
(d) intersect at the vertex

Ans. (a, c)

The circles on focal radii of a parabola as diameter touch:
(a) the tangent at the vertex
(b) the axis
(c) the directrix

Ans. (a)