The general equation to a system of parallel chords in the parabola y 2 = 25 7 x is 4 x - y + k = 0 . What is the equation to the corresponding diameter?

The equation of the parabola is given as y 2 = 25 7 x … . i and the equation of system of parallel chords is 4 x - y + k = 0 … . ( i i ) To solve equations i and i i , we put the value of x from i i in i , y 2 = 25 7 y - k 4 Or 28 y 2 - 25 y + 25 k = 0 Let ordinates of the point of intersection of i and i i be y 1 and y 2 , respectively. Then y 1 and y 2 are the roots of equation i i i . So, y 1 + y 2 = 25 28 The ordinate of the mid-point of the line joining the points whose ordinates are y 1 and y 2 is y 1 + y 2 2 = 25 28 × 2 = 25 56 As the diameter of i corresponding to i i is a line parallel to the axis of i i.e., y = 0 , any chord parallel to i i will be passing through the point whose ordinate is 25 56 . Hence, its equation is y = 25 56 .

The general equation to a system of parallel chords in the parabola y 2 = 25 7 x is 4 x - y + k = 0 . What is the equation to the corresponding diameter?

The equation of the parabola is given as y 2 = 25 7 x … . i and the equation of system of parallel chords is 4 x - y + k = 0 … . ( i i ) To solve equations i and i i , we put the value of x from i i in i , y 2 = 25 7 y - k 4 Or 28 y 2 - 25 y + 25 k = 0 Let ordinates of the point of intersection of i and i i be y 1 and y 2 , respectively. Then y 1 and y 2 are the roots of equation i i i . So, y 1 + y 2 = 25 28 The ordinate of the mid-point of the line joining the points whose ordinates are y 1 and y 2 is y 1 + y 2 2 = 25 28 × 2 = 25 56 As the diameter of i corresponding to i i is a line parallel to the axis of i i.e., y = 0 , any chord parallel to i i will be passing through the point whose ordinate is 25 56 . Hence, its equation is y = 25 56 .

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JEE Advanced

IMPORTANT

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The general equation to a system of parallel chords in the parabola $y^{2} = (\frac{25}{7}) x$ is $4 x - y + k = 0$ . What is the equation to the corresponding diameter?

Important Questions on Conic Sections. The Parabola

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVII>Q 6

$P, Q$ and $R$ are three points on a parabola and the chord $P Q$ cuts the diameter through $R$ in $V$ . Ordinates $P M$ and $Q N$ are drawn to this diameter. Prove that $(i)$ .

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVII>Q 7

Two equal parabolas with axis in opposite directions touch at a point

O

. From a point

P

on one of them, tangents

P Q

and

P Q^{'}

are drawn to the other. Prove that,

Q Q^{'}

will touch the first parabola in

P^{'}

where,

P P^{'}

is parallel to the common tangent at

O

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 1

ω

be the angle which a focal chord of a parabola makes with the axis, prove that the length of the chord is

|4 a {cosec}^{2} ω|

and that the perpendicular on it from the vertex is

|asin ω|

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 2

A point on a parabola, the foot of the perpendicular from it upon the directrix, and the focus are the vertices of an equilateral triangle. Prove that the focal distance of the point is equal to the latus rectum.

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 3

Prove that the semi-latus rectum is a harmonic mean between the segments of any focal chord of a parabola.

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 4

T

be any point on the tangent at any point

P

of a parabola,

T L

be perpendicular to the focal radius

S P

and

T N

be perpendicular to the directrix, then prove that

S L = T N

HARD

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 5

Prove that on the axis of any parabola, there is a certain point

K

which has the property that, if a chord

P Q

of the parabola is drawn through it, then

\frac{1}{P K^{2}} + \frac{1}{Q K^{2}}

is the same for all positions of the chord.

MEDIUM

JEE Advanced

IMPORTANT

The Elements of COORDINATE GEOMETRY Part 1 Cartesian Coordinates>Chapter 8 - Conic Sections. The Parabola>EXAMPLES XXVIII>Q 6

For the parabola

y^{2} = 4 a x

, the normal at the point

(a t_{1}^{2}, 2 a t_{1})

meets the parabola again in the point

(a t_{2}^{2}, 2 a t_{2})

, prove that

t_{2} = - t_{1} - \frac{2}{t_{1}}

The general equation to a system of parallel chords in the parabola y2=257x is 4x-y+k=0. What is the equation to the corresponding diameter?

Important Questions on Conic Sections. The Parabola

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The general equation to a system of parallel chords in the parabola $y^{2} = (\frac{25}{7}) x$ is $4 x - y + k = 0$ . What is the equation to the corresponding diameter?