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JEE Main/Advance

IMPORTANT

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Find the vertex, axis, focus, directrix, latus rectum of the parabola $x^{2} + 2 y - 3 x + 5 = 0 .$

Important Questions on Parabola

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 17

Prove that the locus of the middle points of all tangents drawn from points on the directrix to the parabola

y^{2} = 4 a x

y^{2} (2 x + a) = a {(3 x + a)}^{2}

HARD

JEE Main/Advance

IMPORTANT

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

A variable chord

P Q

of the parabola

y^{2} = 4 x

is drawn parallel to the line

y = x

. If the parameters of the points

P & Q

on the parabola are

p & q

respectively, show that

p + q = 2

. Also show that the locus of the point of intersection of the normals at

P & Q

= 4 A = 4 \times \frac{1}{2} = 2

MEDIUM

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 19

If from the vertex of a parabola a pair of chords be drawn at right angles to one another, & with these chords as adjacent sides a rectangle be constructed, then find the locus of the outer corner of the rectangle.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 20

Two perpendicular straight lines through the focus of the parabola

y^{2} = 4 a x

meet its directrix in

T & T^{'}

respectively. Show that the tangents to the parabola parallel to the perpendicular lines intersect in the mid point of

T T^{'}

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 21

Find the condition on

' a' & b'

so that the two tangents drawn to the parabola

y^{2} = 4 a x

from a point are normals to the parabola

x^{2} = 4 b y .

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 22

T P

and

T Q

are tangents to the parabola and the normal at

P

and

Q

meet at a point

R

on the curve. Prove that the centre of the circle circumscribing the triangle

T P Q

lies on the parabola

2 y^{2} = a (x - a) .

MEDIUM

JEE Main/Advance

IMPORTANT

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

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.

HARD

JEE Main/Advance

IMPORTANT

Beta Question Bank for Engineering: Mathematics>Chapter 18 - Parabola>EXERCISE-4>Q 25

(x_{1}, y_{1}), (x_{2}, y_{2})

and

(x_{3}, y_{3})

be three points on the parabola

y^{2} = 4 a x

and the normals at these points meet in a point, then prove that

\frac{x_{1} - x_{2}}{y_{3}} + \frac{x_{2} - x_{3}}{y_{1}} + \frac{x_{3} - x_{1}}{y_{2}} = 0

Find the vertex, axis, focus, directrix, latus rectum of the parabola x2+2y-3x+5=0.

Important Questions on Parabola

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Find the vertex, axis, focus, directrix, latus rectum of the parabola $x^{2} + 2 y - 3 x + 5 = 0 .$