Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 66

HARD

JEE Main

IMPORTANT

Earn 100

Tangents are drawn from a variable point $P$ to the parabola $y^{2} = 4 a x$ such that they form a triangle of constant area $c^{2}$ with the tangent at the vertex. Show that the locus of $P$ is $x^{2} (y^{2} - 4 a x) = 4 c^{4} a^{2} .$

Important Questions on Conic Section

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 67

If tangents are drawn from points on the line

x = c

to the parabola

y^{2} = 4 a x,

show that the locus of intersection of the corresponding normals is the parabola

a y^{2} = c^{2} (x + c - 2 a)

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 68

If tangents are drawn to the parabola

y^{2} = 4 a x

from a point

T

and the corresponding normals meet in

4 c^{2} a^{2} = x^{2} (y^{2} - 4 a x)

such that

T N

cuts the axis at a fixed point

M

within the curve at a distance

k

from the vertex, show that the locus of

P

is the circle

$x^{2} + y^{2} - (a + k) x + a (2 a - k) = 0$

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 69

Show that the portion of the tangent to a parabola cut off between the directrix and the curve subtends a right angle at the focus.

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 70

Equilateral triangles are circumscribed to the parabola

y^{2} = 4 a x .

Prove that their angular points lie on the conic

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

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 71

If the tangents at the points

P

and

Q

on the parabola

y^{2} = 4 a x

meet at

R

and

S

is its focus, prove that

S R^{2} = S P . S Q .

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 72

If the line

l x + m y + n a = 0

meets the parabola

y^{2} = 4 a x

in

P, Q

and if the lines joining

P, Q

to the focus meet the parabola in

R, T,

show that the equation of

R T

is

a m^{2} - \ln = 0

HARD

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 73

If the line

l x + m y + n a = 0

meets the parabola

y^{2} = 4 a x

in

P, Q

and if the lines joining

P, Q

to the focus meet the parabola in

R, T,

show that the equation of

R T

is

n x - m y + l a = 0

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 74

A rod of length

2 l

slides with its ends on the parabola

y^{2} = 4 a x .

Prove that the midpoint of the rod traces the curve

$(4 a x - y^{2}) (y^{2} + 4 a^{2}) = 4 a^{2} l^{2}$