Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 51

MEDIUM

JEE Main

IMPORTANT

Earn 100

Show that the locus of the foot of the perpendicular from the focus on the tangent at any point of the parabola is the tangent at the vertex.

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 52

Show that the locus of the points of intersection of tangents to

y^{2} = 4 a x

which intercept a constant length

d

on the directrix is

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

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 53

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

is

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

.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 54

If a tangent to the parabola

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

be at right angles to a tangent to the parabola

y^{2} = 4 b (x + b),

show that the locus of the point of intersection of the tangents is the line

x + a + b = 0,

which is the common chord of the two parabolas.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 55

Two lines are drawn at right angles, one being a tangent to

y^{2} = 4 a x

and the other to

x^{2} = 4 b y

. Show that the locus of their point of intersection is the curve

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

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 56

Find the locus of a point such that two of the three normals drawn from it to the parabola

y^{2} = 4 a x

are perpendicular.

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 57

If a chord

P Q

of the parabola

y^{2} = 4 a x

subtends a right angle at the vertex, show that the locus of the point of intersection of normals at

P

and

Q

is

y^{2} = 16 a (x - 6 a)

MEDIUM

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 58

The tangents at

P

and

Q

on

y^{2} = 4 a x

intersect at

T

. The normals at

P

and

Q

intersect at

R

on the curve. Show that the locus of the centre of the circumcircle of

Δ T P Q

is the curve

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

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 59

Find the locus of the point of intersection of those normals to the parabola

x^{2} = 8 y

which are at right angles to each other.