HARD

JEE Main

IMPORTANT

Earn 100

Show that if $r_{1}$ and $r_{2}$ be the lengths of perpendicular chords of a parabola drawn through the vertex, then

${(r_{1} r_{2})}^{\frac{4}{3}} = 16 a^{2} (r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}})$

Important Questions on Conic Section

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 32

Show that the equation of the circle described on the chord intercepted by the parabola

y^{2} = 4 a x

on the line

y = m x + c

as diameter is

$m^{2} (x^{2} + y^{2}) + 2 x (m c - 2 a) - 4 a m y + 4 a m c + c^{2} = 0 .$

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 33

The tangents to the parabola

y^{2} = 4 a x

meet on the parabola

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

Prove that the normals at the points of contact intersect on the tangent at the vertex to the first parabola.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 34

Tangent is drawn at any point

(x_{1}, y_{1})

on the parabola

y^{2} = 4 a x .

Now tangents are drawn from any point on this tangent to the circle

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

such that all the chords of contact pass through a fixed point

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

Prove that

{(\frac{y_{1}}{y_{2}})}^{2} = - 4 (\frac{x_{1}}{x_{2}})

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 35

Find the points on the

x

-axis from which exactly three distinct chords (secants) of the circle

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

can be drawn which are bisected by the parabola

y^{2} = 4 a x, a > 0 .

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 36

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

P

which has the property that if a chord

A B

of the parabola, be drawn through it, then

\frac{1}{A P^{2}} + \frac{1}{B P^{2}}

is the same for all positions of the chord.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 37

Let

P

be a point on the parabola

y^{2} = 4 x,

with the ordinate

y

satisfying

1 < y \leq 2 .

The normal to the parabola at

P

intersects the

x

-axis in

N

and a line parallel to

y

-axis through

P

intersects the

x

-axis in

M

. If

S

be the focus of the parabola and

z =

area of

Δ P M N -

area of

Δ P M S,

find the maximum value of

z .

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 38

If a chord of the parabola

y^{2} = 4 a x

touches the parabola

y^{2} = 4 b x,

then show that the tangent at the extremities of the chord meet on the parabola

b y^{2} = 4 a^{2} x

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 39

P Q

is a chord of a parabola normal at

P

A

is the vertex and through

P

, a line is drawn parallel to

A Q

meeting the axis in

R

. Show that

A R

is double the focal distance of

P .

Show that if r1 and r2 be the lengths of perpendicular chords of a parabola drawn through the vertex, then r1r243=16a2r123+r223

Important Questions on Conic Section

Learn and Prepare for any exam you want !

Show that if $r_{1}$ and $r_{2}$ be the lengths of perpendicular chords of a parabola drawn through the vertex, then

${(r_{1} r_{2})}^{\frac{4}{3}} = 16 a^{2} (r_{1}^{\frac{2}{3}} + r_{2}^{\frac{2}{3}})$