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

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)

The polar focus of parabola

(a) *x-*axis

(b) *y-*axis

(c) Directrix

(d) Latus rectum

The length of the chord of the parabola, y^{2} = 12x passing through the vertex & making an angle of 60^{o} with the axis of x is:

(a) 8

(b) 4

(c) 16/3

The point of intersection of the latus rectum and axis of the parabola y^{2} + 4x + 2y – 8 = 0

(a) (5/4, –1)

(b) (9/4, –1)

(c) (7/2, 5/2)

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

The focal distance of a point on the parabola y^{2} = 16x whose ordinate is twice the abscissa, is

(a) 6

(b) 8

(c) 10

(d) 12

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

The locus of the point of intersection of the perpendicular tangents to the parabola x^{2} = 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

Related: Laws of Exponents questions and answers

An equation of a tangent common to the parabolas y^{2} = 4x and x^{2} = 4y is

(a) x – y + 1 = 0

(b) x + y – 1 = 0

(c) x + y + 1 = 0

(d) y = 0

The ends of the latus rectum of the conic x^{2} + 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)

AP & BP are tangents to the parabola, y^{2} = 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) y^{2} = 2 (x – 1)

The H.M. of the segments of a focal chord of the parabola y^{2} = 4ax is

(a) 4*a*

(b) 2*a*

(c) *a*

(d) *a*^{2}

The equation of the normal to the parabola, y^{2} = 4ax at its point (am^{2}, 2 am) is:

(a) y = – mx + 2am + am^{3}

(b) y = mx – 2am – am^{3
}(c) y = mx + 2am + am^{3}

(d) none

Related: questions on Lines and Angles

The two parabolas y^{2} = 4x and x^{2} = 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

At what point on the parabola y^{2} = 4x the normal makes equal angles with the axes?

(a) (4, 4)

(b) (9, 6)

(c) (4, – 1)

(d) (1, 2)

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

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

If the line y = 2x + k is a tangent to the curve x^{2} = 4y, then *k* is equal to

(a) 4

(b) ½

(c) –4**
**(d) –1/2

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.

Two perpendicular tangents to y^{2} = ax always intersect on the line, if

(a) x = a

(b) x + a = 0

(c) x + 2a = 0

(d) x + 4a = 0

If the distances of two points P & Q from the focus of a parabola y^{2} = 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

PQ is any focal chord of the parabola y^{2} = 32x. The length of PQ can never be less than

(a) 8 units

(b) 16 units

(c) 32 units

(d) 48 units

From the point (4, 6) a pair of tangent lines are drawn to the parabola, y^{2} = 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

Related: Euclid’s geometry questions

The tangent drawn at any point *P* to the parabola y^{2} = 4ax meets the directrix at point *K*, then the angle which *KP* subtends at its focus is

(a) 30^{o}

(b) 45^{o
}(c) 60^{o}

(d) 90^{o}

Locus of the intersection of the tangents at the ends of the normal chords of the parabola

y^{2} = 4ax is

(a) (2a + x) y^{2} + 4a^{3} = 0

(b) (2a + x) + y^{2} = 0

(c) (2a + x) y^{2} + 4a = 0

(d) none of these

The length intercepted by the curve y^{2} = 4x on the line satisfying dy/dx = 1 and passing through point (0, 1) is given by

(a) 1

(b) 2

(c) 0

Tangents are drawn from the points on the line x – y + 3 = 0 to parabola y^{2} = 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)

The points of intersection of the curves whose parametric equations are x = t^{2} + 1, y = 2t and x = 2s, y = 2/s is given by

(a) (1, –3)

(b) (2, 2)

(c) (–2, 4)

(d) (1, 2)

Related: Mensuration aptitude questions

The distance between a tangent to the parabola y^{2} = 4 A x (A > 0) and the parallel normal with gradient 1 is:

(a) 4 A

(b) 2 A

(c) 2 A

(d) A

The angle between two curves y^{2} = 4(x + 1) and x^{2} = 4(y + 1) is

(a) 0^{o}

(b) 90^{o
}(c) 60^{o}

(d) 30^{o}

AB is a chord of the parabola y^{2} = 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

The length of the latus rectum of the parabola x = ay^{2} + by + c is

(a) a/4

(b) a/3

(c) 1/a

(d) 1/4a

T is a point on the tangent to a parabola y^{2} = 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)

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

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

The equation of the other normal to the parabola y^{2} = 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

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) y^{2} – 2ax + a^{2} = 0

(b) y^{2} + 2ax + a^{2} = 0

(c) x^{2} – 2ay + a^{2} = 0

(d) x^{2} + 2ay + a^{2} = 0

Related: Percentages problems

If x + y = k, is the normal to y^{2} = 12x, then k is

(a) 3

(b) 9

(c) –9

(d) – 3

A set of parallel chords of the parabola y^{2} = 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

The focal chord to y^{2} = 16 x is tangent to (x – 6)^{2} + y^{2} = 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}

The order of the differential equation of all parabolas having directrix parallel to *x*-axis is

(a) 3

(b) 1

(c) 4

(d) 2

Related: mcq on Compound Interest

Let P be the point (1, 0) and Q a point of the locus y^{2} = 8x. The locus of mid point of PQ is

(a) x^{2} + 4y + 2 = 0

(b) x^{2} – 4y + 2 = 0

(c) y^{2} – 4x + 2 = 0

(d) y^{2} + 4x + 2 = 0

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

If the normals at two points *P* and *Q* of a parabola y^{2} = 4ax intersect at a third point *R* on the curve, then the product of ordinates of *P* and *Q* is

(a) 4a^{2}

(b) 2a^{2
}(c) –4a^{2}

(d) 8a^{2}

A circle touch the line L and the circle C_{1} 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

The parametric equation of the curve y^{2} = 8x are

(a) x = t^{2}, y = 2t

(b) x = 2t^{2}, y = 4t

(c) x = 2t, y = 4t^{2}

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) x^{2} y + 2 = 0

(b) xy + 2 = 0

(c) xy – 2 = 0

(d) x^{2} + xy – y – 1 = 0

The centroid of the triangle formed by joining the feet of the normals drawn from any point to the parabola y^{2} = 4ax, lies on

(a) Axis

(b) Directrix

(c) Latusrectum

(d) Tangent at vertex

P is a point on the parabola y^{2} = 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)

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

The circles on focal radii of a parabola as diameter touch:

(a) the tangent at the vertex

(b) the axis

(c) the directrix