HARD

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From a point $P (α, β)$ , three real and distinct normals are drawn to the parabola $y^{2} = 8 x$ . Let $A, B$ and $C$ be a feet of normals. The normals at $A, B$ and $C$ cut the $x$ -axis at $Q$ , $R$ and $S$ respectively. The value of $O Q + O R + O S$ ( $O$ is vertex) must be greater than a positive integer $k$ , find the greatest value of $k$ .

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Important Questions on Parabola

HARD

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

Let the circles

C_{1} : x^{2} + y^{2} = 9

and

C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16,

intersect at the points

X

and

Y .

Suppose that another circle

C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = r^{2}

satisfies the following conditions:

(i)

centre of

C_{3}

is collinear with the centres of

C_{1}

and

C_{2}

(i i) C_{1}

and

C_{2}

both lie inside

C_{3},

and

(i i i) C_{3}

touches

C_{1}

M

and

C_{2}

N .

Let the line through

X

and

Y

intersect

C_{3}

Z

and

W,

and let a common tangent of

C_{1}

and

C_{3}

be a tangent to the parabola

x^{2} = 8 α y .

There are some expression given in the List-

I

whose values are given in List-

I I

below:

	List- $I$		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

Which of the following is the only INCORRECT combination?

HARD

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

y = m x + 4

is a tangent to both the parabolas,

y^{2} = 4 x

and

x^{2} = 2 b y,

then

b

is equal to

HARD

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

Let the circles

C_{1} : x^{2} + y^{2} = 9

and

C_{2} : {(x - 3)}^{2} + {(y - 4)}^{2} = 16,

intersect at the points

X

and

Y .

Suppose that another circle

C_{3} : {(x - h)}^{2} + {(y - k)}^{2} = r^{2}

satisfies the following conditions:

(i)

centre of

C_{3}

is collinear with the centres of

C_{1}

and

C_{2}

(i i) C_{1}

and

C_{2}

both lie inside

C_{3},

and

(i i i) C_{3}

touches

C_{1}

M

and

C_{2}

N .

Let the line through

X

and

Y

intersect

C_{3}

Z

and

W,

and let a common tangent of

C_{1}

and

C_{3}

be a tangent to the parabola

x^{2} = 8 α y .

There are some expression given in the List-

I

whose values are given in List-

I I

below:

	List- I		List- II
$(I)$	$2 h + k$	$(P)$	$6$
$(I I)$	$\frac{L e n g t h o f Z W}{L e n g t h o f X Y}$	$(Q)$	$\sqrt{6}$
$(I I I)$	$\frac{A r e a o f t r i a n g l e M Z N}{A r e a o f t r i a n g l e Z M W}$	$(R)$	$\frac{5}{4}$
$(I V)$	$α$	$(S)$	$\frac{21}{5}$
		$(T)$	$2 \sqrt{6}$
		$(U)$	$\frac{10}{3}$

Which of the following is the only INCORRECT combination?

MEDIUM

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

Let

a, b \in R

and

a > 0 .

If the tangent at the point

(2, 2)

to the circle

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

touches the parabola,

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

then

b - a

is equal to_______.

HARD

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

Tangent and normal are drawn at

P (16, 16)

on the parabola

y^{2} = 16 x

, which intersect the axis of the parabola at

A & B

, respectively. If

C

is the center of the circle through the points

P, A & B

and

∠ C P B = θ,

then a value of

\tan θ

is:

HARD

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

Let

L_{1}

be a tangent to the parabola

y^{2} = 4 (x + 1)

and

L_{2}

be a tangent to the parabola

y^{2} = 8 (x + 2)

such that

L_{1}

and

L_{2}

intersect at right angles. Then

L_{1}

and

L_{2}

meet on the straight line:

MEDIUM

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

If one end of a focal chord

A B

of the parabola

y^{2} = 8 x

is at

A (\frac{1}{2}, - 2),

then the equation of the tangent to it at

B

is:

HARD

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

The angle between the tangents drawn from the point

(1, 4)

to the parabola

y^{2} = 4 x

HARD

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

Columns 1, 2 and 3 contain conics, equations of tangents to the conics and points of contact, respectively.

Column 1	Column 2	Column 3
(I) $x^{2} + y^{2} = a^{2}$	(i) $m y = m^{2} x + a$	(P) $(\frac{a}{m^{2}}, \frac{2 a}{m})$
(II) $x^{2} + a^{2} y^{2} = a^{2}$	(ii) $y = m x + a \sqrt{m^{2} + 1}$	(Q) $(\frac{- m a}{\sqrt{m^{2} + 1}}, \frac{a}{\sqrt{m^{2} + 1}})$
(III) $y^{2} = 4 a x$	(iii) $y = m x + \sqrt{a^{2} m^{2} - 1}$	(R) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} + 1}}, \frac{1}{\sqrt{a^{2} m^{2} + 1}})$
(IV) $x^{2} - a^{2} y^{2} = a^{2}$	(iv) $y = m x + \sqrt{a^{2} m^{2} + 1}$	(S) $(\frac{- a^{2} m}{\sqrt{a^{2} m^{2} - 1}}, \frac{- 1}{\sqrt{a^{2} m^{2} - 1}})$

If a tangent to a suitable conic (Column 1) is found to be

y = x + 8

and its point of contact is

(8, 16)

, then which of the following options is the only Correct combination?

MEDIUM

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

The shortest distance between the line

y = x

and the curve

y^{2} = x - 2

MEDIUM

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

The shortest distance between the point

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

and the curve

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

, is

HARD

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

If the tangent to the conic,

y - 6 = x^{2}

(2, 10)

touches the circle,

x^{2} + y^{2} + 8 x - 2 y = k

(for some fixed

k

) at a point

(α, β);

then

(α, β)

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal 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

HARD

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

The equation of a tangent to the parabola,

x^{2} = 8 y

, which makes an angle

θ

with the positive direction of

x -

axis, is

HARD

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

The shortest distance between the parabolas

y^{2} = 4 x

and

y^{2} = 2 x - 6

MEDIUM

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

x - 2 y + k = 0

is a tangent to the parabola

y^{2} - 4 x - 4 y + 8 = 0,

then the slope of the tangent drawn at

(1, k)

on the given parabola is

MEDIUM

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

The number of points on the parabola

y^{2} = x

at which the slope of the normal drawn at the point is equal to the

x

-coordinate of that point is

EASY

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

Let a line

y = m x (m > 0)

intersect the parabola,

y^{2} = x

at a point

P

other than the origin. Let the tangent to it at

P

meet the

x

-axis at the point

Q .

If area

(Δ O P Q) = 4

square units, then

m

is equal to

MEDIUM

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

The slope of the line touching both the parabolas

y^{2} = 4 x

and

x^{2} = - 32 y

HARD

Mathematics>Coordinate Geometry>Parabola>Tangent and Normal 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 -

Important Questions on Parabola

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