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If tangents are drawn from a point $P$ to the parabola $y^{2} = 4 x$ with focus $(S)$ such that the lengths of the tangents are $\sqrt{5}$ and $\sqrt{10},$ then $S P$ is equal to

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Important Questions on Parabola

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

a, b

and

c

be the

7^{t h}, 11^{t h}

and

13^{t h}

terms respectively of a non-constant A.P. If these are also the three consecutive terms of a G.P., then

\frac{a}{c}

is equal to:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

a, b

and

c

be three distinct real numbers in G.P. and

a + b + c = x b,

then

x

cannot be:

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

b_{i} > 1

for

i = 1, 2, \dots ., 101 .

Suppose

\log_{e} b_{1}, \log_{e} b_{2}, \dots . ., \log_{e} b_{101}

are in Arithmetic Progression (A.P.) with the common difference

\log_{e} 2 .

Suppose

a_{1}, a_{2}, \dots ., a_{101}

are in A.P. such that

a_{1} = b_{1}

and

a_{51} = b_{51} .

t = b_{1} + b_{2} + \dots + b_{51}

and

s = a_{1} + a_{2} + \dots + a_{51}

, then

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The sum of the

3^{r d}

and the

4^{t h}

terms of a

G . P .

60

and the product of its first three terms is

1000.

If the first term of this

G . P .

is positive, then its

7^{t h}

term is:

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The focus of the parabola

y = 2 x^{2} + x

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The sum of first

20

terms of the sequence

0.7, 0.77, 0.777, . . . . . .,

is :

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If $m$ is the $A . M .$ of two distinct real numbers $I$ and $n$ $(I, n > 1)$ and $G_{1}, G_{2} and G_{3}$ are three geometric means between $I and n$ , then $G_{1}^{4} + 2 G_{2}^{4} + G_{3}^{4}$ equals

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The sum of the first

20

terms of the series

1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If the tangents and normals at the extremities of a focal chord of a parabola intersect at

(x_{1}, y_{1})

and

(x_{2}, y_{2})

respectively, then

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

E

denote the parabola

y^{2} = 8 x .

Let

P = (- 2, 4)

, and let

Q

and

Q^{'}

be two distinct points on

E

such that the lines

P Q

and

P Q^{'}

are tangents to

E

. Let

F

be the focus of

E

. Then which of the following statements is (are) TRUE?

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Three positive numbers form an increasing

G . P .

If the middle term in this

G . P .

is doubled, the new numbers are in

A . P .

Then the common ratio of the

G . P .

is :

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

A (1, 2), B (4, - 4), C (2, 2 \sqrt{2})

be points on the parabola

y^{2} = 4 x .

α

and

β

respectively represent the area of

Δ A B C

and the area of the triangle formed by the tangents at

A, B, C

to the above parabola, then

α β =

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The quotient when

1 + x^{2} + x^{4} + . . . + x^{34}

is divided by

1 + x + x^{2} + . . . + x^{17}

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The number of real solutions of the equation

\sin^{- 1} (\sum_{i = 1}^{\infty} x^{i + 1} - x \sum_{i = 1}^{\infty} {(\frac{x}{2})}^{i}) = \frac{π}{2} - \cos^{- 1} (\sum_{i = 1}^{\infty} {(- \frac{x}{2})}^{i} - \sum_{i = 1}^{\infty} {(- x)}^{i})

lying in the interval

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

is____.

(Here, the inverse trigonometric functions

\sin^{- 1} x & \cos^{- 1} x

assume values in

[- \frac{π}{2}, \frac{π}{2}] & [0, π]

respectively.)

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Suppose

p, q, r

are real numbers such that

q = p (4 - p), r = q (4 - q), p = r (4 - r) .

The maximum possible value of

p + q + r

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

f : x \to y

be such that

f (1) = 2

and

f (x + y) = f (x) f (y)

for all natural numbers

x

and

y

.

\sum_{k = 1}^{n} f (a + k) = 16 (2^{n} - 1)

then

a

is equal to

HARD

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Let

x, y, z

be three non-negative integers such that

x + y + z = 10

. The maximum possible value of

x y z + x y + y z + z x

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

The sum of the first

10

terms of the series

9 + 99 + 999 + . \dots .

EASY

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

If the three normals drawn to the parabola,

y^{2} = 2 x

pass through the point

(a, 0), a \neq 0

, then

a

must be greater than :

MEDIUM

Mathematics>Coordinate Geometry>Parabola>Properties of Parabola

Suppose

O A B C

is a rectangle in the

x y

-plane where

O

is the origin and

A, B

lie on the parabola

y = x^{2} .

Then

C

must lie on the curve-

If tangents are drawn from a point P to the parabola y2=4x with focus (S) such that the lengths of the tangents are 5 and 10, then SP is equal to

Important Questions on Parabola

Learn and Prepare for any exam you want !

If tangents are drawn from a point $P$ to the parabola $y^{2} = 4 x$ with focus $(S)$ such that the lengths of the tangents are $\sqrt{5}$ and $\sqrt{10},$ then $S P$ is equal to