HARD

Earn 100

An equilateral triangle $ABC$ is inscribed in the parabola $y = x^{2}$ and one of the sides of the equilateral triangle has the gradient $2$ . If the sum of $x$ -coordinates of all the vertices of the triangle is expressed in the form of $\frac{p}{q}$ (where $p & q$ are coprime natural number), then the value of $|p - q|$ is

56.52% studentsanswered this correctly

Important Questions on Parabola

HARD

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The length of the chord of the parabola

x^{2} = 4 y

having equation

x - \sqrt{2} y + 4 \sqrt{2} = 0

EASY

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

Find the value of

k

if the line

2 y = 5 x + k

is tangent to the parabola

y^{2} = 6 x

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

If the

x

-intercept of a focal chord of the parabola

y^{2} = 8 x + 4 y + 4

3

, then the length of this chord is equal to _____ .

HARD

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

P (h, k)

be point on the parabola

x = 4 y^{2}

, which is nearest to the point

Q (0, 33)

, then the distance of

P

from the directrix of the parabola

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

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The length of the chord of the parabola

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

which passes through the vertex and makes an acute angle

α

with the axis of the parabola is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The distance of the point

(6, - 2 \sqrt{2})

from the common tangent

y = m x + c, m > 0

, of the curves

x = 2 y^{2}

and

x = 1 + y^{2}

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

Let

P_{1}

be a parabola with vertex

(3, 2)

and focus

(4, 4)

and

P_{2}

be its mirror image with respect to the line

x + 2 y = 6

. Then the directrix of

P_{2}

x + 2 y =

_____.

HARD

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

A triangle is formed by the tangents at the point

(2, 2)

on the curves

y^{2} = 2 x

and

x^{2} + y^{2} = 4 x

, and the line

x + y + 2 = 0

. If

r

is the radius of its circumcircle, then

r^{2}

is equal to

EASY

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The line

y = 4 x + c

touches the parabola

y^{2} = 4 x

, if

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

Let

C

be the locus of the mirror image of a point on the parabola

y^{2} = 4 x

with respect to the line

y = x

. Then the equation of tangent to

C

P (2, 1)

is :

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The equations of two sides of a variable triangle are

x = 0

and

y = 3

, and its third side is a tangent to the parabola

y^{2} = 6 x

. The locus of its circumcentre is :

EASY

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The line

y = mx + 2

intersects the parabola

y = {ax}^{2} + 5 x - 2

(1, 5)

. Then the value of

a + m

is equal to

HARD

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The equations of sides

A B

and

A C

of a triangle

A B C

are

(λ + 1) x + λ y = 4

and

λ x + (1 - λ) y + λ = 0

respectively. Its vertex

A

is on the

y -

axis and its orthocentre is

(1, 2)

. The length of the tangent from the point

C

to the part of the parabola

y^{2} = 6 x

in the first quadrant is

HARD

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

Let a tangent to the curve

y^{2} = 24 x

meet the curve

x y = 2

at the points

A

and

B

. Then the mid-points of such line segments

A B

lie on a parabola with the

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

Let

R

be the focus of the parabola

y^{2} = 20 x

and the line

y = m x + c

intersect the parabola at two points

P

and

Q

. Let the points

G (10, 10)

be the centroid of the triangle

P Q R

. If

c - m = 6

, then

P Q^{2}

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

Let the tangent to the curve

x^{2} + 2 x - 4 y + 9 = 0

at the point

P (1, 3)

on it meet the

y -

axis at

A .

Let the line passing through

P

and parallel to the line

x - 3 y = 6

meet the parabola

y^{2} = 4 x

B .

B

lies on the line

2 x - 3 y = 8,

then

{(A B)}^{2}

is equal to

_______.

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

Let

S

be the set of all

a \in N

such that the area of the triangle formed by the tangent at the point

P (b, c), b, c \in N

, on the parabola

y^{2} = 2 a x

and the lines

x = b, y = 0

16 {unit}^{2}

, then

\sum_{a \in S} a

is equal to _____ .

EASY

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The condition that the straight line

l x + m y + n = 0

touches the parabola

x^{2} = 4 a y

HARD

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

Let

P

be a point on the parabola,

y^{2} = 12 x

and

N

be the foot of the perpendicular drawn from

P

, on the axis of the parabola. A line is now drawn through the mid-point

M

P N

, parallel to its axis which meets the parabola at

Q

. If the

y -

intercept of the line

NQ

\frac{4}{3},

then :

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal to a Parabola

The vertices of the base of an isosceles triangle lie on a parabola

y^{2} = 4 x

and the base is a part of the line

y = 2 x - 4

. If the third vertex of the triangle lies on the

x

-axis, its coordinates are

An equilateral triangle ABC is inscribed in the parabola y=x2 and one of the sides of the equilateral triangle has the gradient 2. If the sum of x -coordinates of all the vertices of the triangle is expressed in the form of pq (where p & q are coprime natural number), then the value of p-q is

Important Questions on Parabola

Learn and Prepare for any exam you want !