A and B are two points whose co-ordinates are a t 2 , 2 a t and a t 2 , - 2 a t and S is the point ( a , 0 ) . Prove that 1 S A + 1 S B = 1 a .

Given the coordinates of points A = a t 2 , 2 a t B = a t 2 , - 2 a t and S = a , 0 Distance between two points C x 1 , y 1 a n d D x 2 , y 2 is given by C D = x 2 - x 1 2 + y 2 - y 1 2 So, from distance formula S A = a t 2 - a 2 + 2 a t - 0 2 = a t 4 - 2 t 2 + 1 + 4 t 2 = a t 2 + 1 Similarly, S B = a t 2 - a 2 + - 2 a t - 0 2 = a 1 t 4 - 2 t 2 + 1 + 4 t 2 = a 1 t 2 + 1 2 = a 1 t 2 + 1 So, 1 S A + 1 S B = 1 a t 2 + 1 + 1 a 1 t 2 + 1 = 1 a t 2 + 1 + t 2 a t 2 + 1 = 1 + t 2 a 1 + t 2 = 1 a

HARD

JEE Main/Advanced

IMPORTANT

Earn 100

Show that the centroid ' $G$ ' of a triangle divides the join of its orthocentre ' $P$ ' circumcentre ' $O$ ' in the ratio $2 : 1$ .

Important Questions on Coordinate Systems and Coordinates

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 1 - Coordinate Systems and Coordinates>EXERCISE ON LEVEL-I>Q 13

A

and

B

are two points whose co-ordinates are

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

and

(\frac{a}{t^{2}}, - \frac{2 a}{t})

and

S

is the point

(a, 0)

. Prove that

\frac{1}{S A} + \frac{1}{S B} = \frac{1}{a}

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 1 - Coordinate Systems and Coordinates>EXERCISE ON LEVEL-I>Q 14

The line joining

A (b cosα, b sinα)

and

B (a cosβ, a sinβ)

is produced to the point

M (x, y)

so that

AM

and

BM

are in the ratio

b : a

. Then prove that

x + y \tan [\frac{α + β}{2}] = 0

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 1 - Coordinate Systems and Coordinates>EXERCISE ON LEVEL-I>Q 15

x_{1}, x_{2}, x_{3}

are in

A . P .

and

y_{1}, y_{2}, y_{3}

are also in

A . P .

, prove that the points

(x_{1}, y_{1}), (x_{2}, y_{2})

(x_{3}, y_{3})

are collinear.

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 1 - Coordinate Systems and Coordinates>EXERCISE ON LEVEL-I>Q 16

A (a_{1}, b_{1}), B (a_{2}, b_{2}), C (a_{3}, b_{3})

are the vertices of a

∆ A B C

. The side

A B

is divided by the point

D

in the ratio

λ : μ

and then the line segment

D C

is divided by the point

E

in the ratio

v : λ + μ

. Find the co-ordinates of

E

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 1 - Coordinate Systems and Coordinates>EXERCISE ON LEVEL-I>Q 17

If the area of the quadrilateral whose angular points taken in order are

(1, 2), (- 5, 6), (7, - 4)

and

(λ, - 2)

be zero, show that

λ - 3 = 0

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 1 - Coordinate Systems and Coordinates>EXERCISE ON LEVEL-I>Q 18

\tan α, \tan β, \tan γ

are the roots of the equation

x^{3} - 3 a x^{2} + 3 b x - 1 = 0

. Find the centroid of the triangle whose vertices are

(\tan α, \cot α), (\tan β, \cot β), (\tan γ, \cot γ) .

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 1 - Coordinate Systems and Coordinates>EXERCISE ON LEVEL-I>Q 19

a, b, c

are distinct real numbers, show that the points

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

and

(c, c^{2})

are not collinear.

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 1 - Coordinate Systems and Coordinates>EXERCISE ON LEVEL-I>Q 20

A (\cos θ, \sin θ), B (\sin θ, - \cos θ), C (1, 2)

are the vertices of a

Δ A B C

. Find the locus of its centroid if

θ

varies.

Show that the centroid ' G ' of a triangle divides the join of its orthocentre ' P ' circumcentre ' O ' in the ratio 2: 1.

Important Questions on Coordinate Systems and Coordinates

Learn and Prepare for any exam you want !

Show that the centroid ' $G$ ' of a triangle divides the join of its orthocentre ' $P$ ' circumcentre ' $O$ ' in the ratio $2 : 1$ .