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A chord through the point $(1, - 2)$ cuts the curve $3 x^{2} - y^{2} - 2 x + 4 y = 0$ at $P$ and $Q .$ If $P Q$ subtends an angle $' θ'$ at the origin, then $θ$ equals $__________$

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Important Questions on Pair of Straight lines

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Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

(p, q)

is the centroid of the triangle formed by the lines

8 x^{2} - 14 x y + 5 y^{2} = 0

and

x - 2 y + 3 = 0

, then

HARD

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

(a + 3 b) (3 a + b) = 4 h^{2}

, then the angle between the lines represented by

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

HARD

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

l x^{2} + 3 x y - 2 y^{2} - 5 x + 5 y + k = 0

represents a pair of perpendicular lines, then

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

The ratio in which the line

x + y - 1 = 0

divides the line segment joining the origin and the point of intersection of the lines represented by

2 x^{2} - 13 x y - 7 y^{2} + x + 23 y - 6 = 0

EASY

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

θ

is the angle between the pair of straight lines

x^{2} - 5 x y + 4 y^{2} + 3 x - 4 = 0

,

then

\tan^{2} θ

is equal to

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

If the pair of lines $6 x^{2} + x y - y^{2} = 0$ and $3 x^{2} - a x y - y^{2} = 0, a > 0$ have a common line. then $a =$

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

The sine of the angle between the pair of lines represented by the equation

x^{2} - 7 x y + 12 y^{2} = 0

EASY

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

The auxiliary equation of the lines passing through the origin and having slopes

\sqrt{3} + 1

and

\sqrt{3} - 1

EASY

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

x^{2} + y^{2} + 2 g x + 2 f y + 1 = 0

represents a pair of straight lines, then

f^{2} + g^{2}

is equal to

EASY

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

The value of '

p

' for which the equation

x^{2} + p x y + y^{2} - 5 x - 7 y + 6 = 0

represents a pair of straight lines, is

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

If one of the lines given by the equation

x^{2} + k x y + 2 y^{2} = 0

x + 2 y = 0,

then

k =

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

If slopes of lines represented by

K x^{2} + 5 x y + y^{2} = 0

differ by

1

then

K =

HARD

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

The joint equation of a pair of lines passing through the point of intersection of lines

2 x^{2} - x y - 15 y^{2} - 7 x + 32 y - 9 = 0

and parallel to co-ordinate axes is

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

A pair of straight lines is passing through the point

(1, 1)

. One of the lines makes an angle

θ

with the positive direction of

X

-axis and the other makes the same angle with the positive direction of

Y

-axis. If the equation of the pair of straight lines is

x^{2} - (a + 2) x y + y^{2} + a (x + y - 1) = 0, a \neq - 2

, then the value of

θ

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

If the equation

x^{2} - 3 x y + λ y^{2} + 3 x - 5 y + 2 = 0

represents a pair of lines, where

λ

is a real number and

θ

is an angle between them, then value of

cosec {}^{2}θ

HARD

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

The product of lengths of the perpendiculars drawn from the point

(- 1, 5)

to the pair of lines

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

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

The number of values of

a

for which the pair of lines represented by

3 a x^{2} + 5 x y + (a^{2} - 2) y^{2} = 0

are at right angles to each other, is

HARD

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

If the slope of one of the lines represented by

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

be the square of the other, then

\frac{a + b}{h} + \frac{8 h^{2}}{a b}

MEDIUM

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

If lines represented by equation

p x^{2} - q y^{2} = 0

are distinct then

EASY

Mathematics>Coordinate Geometry>Pair of Straight lines>Equations of Pair of Lines

The normal to the curve

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

, at

(1, 1)

A chord through the point (1, -2) cuts the curve 3x2-y2-2x+4y=0 at P and Q. If PQ subtends an angle 'θ' at the origin, then θ equals __________

Important Questions on Pair of Straight lines

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A chord through the point $(1, - 2)$ cuts the curve $3 x^{2} - y^{2} - 2 x + 4 y = 0$ at $P$ and $Q .$ If $P Q$ subtends an angle $' θ'$ at the origin, then $θ$ equals $__________$