MEDIUM

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Consider the curves $S_{1} \equiv y^{2} - 4 k_{1} x = 0, k_{1} > 0, S_{2} \equiv x^{2} - 4 k_{2} y = 0, k_{2} > 0$ and $S_{3} \equiv x^{2} + y^{2} + 10 y - 27 = 0$ . A line $L \equiv 2 y + 3 x + 36 = 0$ touches $S_{3} = 0, S_{1} = 0 & S_{2} = 0$ $P, Q$ and $R$ respectively such that $P Q = P R = \sqrt{1053}$ . Moreover $S_{1}$ and $S_{2}$ intersect at point $S$ (other than origin)

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Important Questions on Parabola

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals 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

_______.

HARD

Mathematics>Coordinate Geometry>Parabola>Normals 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

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals 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}

HARD

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

P

and

Q

are two distinct points on the parabola,

y^{2} = 4 x

, with parameters

t

and

t_{1}

, respectively. If the normal at

P

passes through

Q

, then the minimum value of

t_{1}^{2}

, is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals 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>Normals to a Parabola

The shortest distance between the point

(\frac{3}{2}, 0)

and the curve

y = \sqrt{x}, (x > 0)

, is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals 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>Normals to a Parabola

The normal at a point on the parabola

y^{2} = 4 x

passes through

(5, 0)

. If two more normals to this parabola also pass through

(5, 0),

then the centroid of the triangle formed by the feet of these three normals is

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals 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 :

EASY

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

The condition that the straight line

l x + m y + n = 0

touches the parabola

x^{2} = 4 a y

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

Normals drawn to

y^{2} = 4 a x

at the points where it is intersected by the line

y = m x + c

intersected at

P

. Coordinates of foot of the another normal drawn to the parabola from the point

P

EASY

Mathematics>Coordinate Geometry>Parabola>Normals 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>Normals to a Parabola

The shortest distance between the parabolas

y^{2} = 4 x

and

y^{2} = 2 x - 6

HARD

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

Let

P

be the point on the parabola

y^{2} = 4 x

which is at the shortest distance from the center

S

of the circle

x^{2} + y^{2} - 4 x - 16 y + 64 = 0 .

Let

Q

be the point on the circle dividing the line segment

S P

internally. Then -

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

Three normals are drawn from the point

(c, 0)

to the curve

y^{2} = x

. If one of the normals is

X

-axis, then the value of

c

for which the other two normals are perpendicular to each other is

EASY

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

The normal to the parabola

y^{2} = 8 x

at the point

(2, 4)

meets the parabola again at the point

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals to a Parabola

The equation of the normal to the parabola

y^{2} = 4 x

which is perpendicular to

x + 3 y + 1 = 0

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Normals 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>Normals 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>Normals 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

Consider the curves S1≡y2-4k1x=0,k1>0,S2≡x2-4k2y=0,k2>0 and S3≡x2+y2+10y-27=0. A line L≡2y+3x+36=0 touches S3=0,S1=0&S2=0 P, Q and R respectively such that PQ=PR=1053. Moreover S1 and S2 intersect at point S (other than origin)

Important Questions on Parabola

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