Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 23

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Let $S$ is the focus of the parabola $y^{2} = 4 a x$ and $X$ the foot of the directrix, $P P'$ is a double ordinate of the curve and $P X$ meets the curve again in $Q .$ Prove that $P' Q$ passes through focus.

Important Questions on Parabola

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Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-1>Q 22

T P &

T Q

are the tangents to the parabola,

y^{2} = 4 a x

at

P

&

Q

. If the chord

P Q

passes through the fixed point

(- a, b)

then the locus of

∴ x = a t^{2} & x = \frac{a}{t^{2}}

is:

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Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-2>Q 1

The straight line joining any point

P

on the parabola

y^{2} = 4 a x

to the vertex and perpendicular from the focus to the tangent at

P

, intersect at

R

, then the equation of the locus of

R

is -

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Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-2>Q 5

The length of the chord of the parabola

y^{2} = x

which is bisected at the point

(2, 1)

is-

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Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-2>Q 8

The locus of the mid point of the focal radii of a variable point moving on the parabola,

y^{2} = 4 a x

is a parabola whose

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Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-2>Q 10

The tangent and normal at $P(t)$, for all real positive t, to the parabola

y^{2} = 4 a x

meet the axis of the parabola in

T

and

G

respectively, then the angle at which the tangent at

P

to the parabola is inclined to the tangent at

P

to the circle through the points

P

,

T

and

G

is -

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Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-2>Q 11

A variable circle is described to pass through the point

(1, 0)

and tangent to the curve

y = \tan (\tan^{- 1} x)

. The locus of the centre of the circle is a parabola whose:

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Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-2>Q 12

A B, A C

are tangents to a parabola

y^{2} = 4 a x

,

p_{1}, p_{2}, p_{3}

are the lengths of the perpendiculars from

A, B, C

on any tangent to the curve, then

p_{2}, p_{1}, p_{3}

are in

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Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 28

If tangent drawn at a point

(t^{2}, 2 t)

on the parabola

y^{2} = 4 x

is same as the normal drawn at a point

(\sqrt{5} \cos ϕ, 2 \sin ϕ)

on the ellipse

4 x^{2} + 5 y^{2} = 20

. Find the values of

t & ϕ

.