Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 63

EASY

JEE Main

IMPORTANT

Earn 100

If the sum of the tangents of the base angles of a triangle described on a given base be constant, show that the locus of its vertex is a parabola.

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 64

Prove that the locus of the centres of equilateral triangles inscribed in the parabola

y^{2} = 4 a x

is

9 y^{2} = 4 a (x - 8 a) .

EASY

JEE Main

IMPORTANT

Eduwiser's Coordinate Geometry for JEE Main and Advanced>Chapter 7 - Conic Section>7.1>Q 65

Prove that the locus of the circumcentre of the variable triangle having sides

x = 0, y = 2

and

l x + m y = 1,

where

(l, m)

lies on the parabola

y^{2} = 4 x

is also a parabola.

HARD

JEE Main

IMPORTANT

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

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} .

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

N

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 .