Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 4

EASY

JEE Main/Advanced

IMPORTANT

Earn 100

Prove that the bisectors of the angles between the lines represented by $a x^{2} + 2 h x y + b y^{2} = λ (x^{2} + y^{2})$ are always the same, whatever be the values of $λ$ .

Important Questions on Pair of Straight Lines

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 5

Show that all those chords of the curve

3 x^{2} - y^{2} - 2 x + 4 y = 0

which subtend a right angle at the origin pass through a fixed point. Find that point.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 6

A triangle has

a x^{2} + 2 h x y + b y^{2} = 0

for two of its sides and the point

(α, β)

for the orthocentre . Prove that the third side is given by

(a + b) (α x + β y) = a β^{2} - 2 h α β + b α^{2}

.

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 7

For what value of

λ

does the equation

6 x^{2} + 13 x y + 6 y^{2} + 8 x + 7 y + λ = 0

represents a pair of straight lines? Show that the angle between them is

\tan^{- 1} (\frac{5}{12})

. Also find the point of intersection of the lines.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 8

Show that the equation

4 x^{2} + 4 x y + y^{2} - 6 x - 3 y - 4 = 0

represents a pair of parallel lines. Find the distance between them.

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 9

Show that the four lines given by the equations

3 x^{2} + 8 x y - 3 y^{2} = 0

and

3 x^{2} + 8 x y - 3 y^{2} + 2 x - 4 y - 1 = 0

form a square. Find equations of the diagonals of the square.

EASY

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 10

Find the value of

c

so that the lines joining the origin to the common points of

{(x - 3)}^{2} + {(y - 4)}^{2} = c^{2}

and

4 x + 3 y = 24

are at right angles.

MEDIUM

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 11

Find the joint equation of pair of lines through the origin and making equilateral triangle with the line

x = 3 .

HARD

JEE Main/Advanced

IMPORTANT

Skills in Mathematics Coordinate Geometry for JEE Main & Advanced>Chapter 3 - Pair of Straight Lines>EXERCISE ON LEVEL-I>Q 12

Show that the lines

(1 - \cos θ \tan α) y^{2} - (2 \cos θ + \sin^{2} θ \tan α) x y + \cos θ (\cos θ + \tan α) x^{2} = 0

include an angle

α

between them.