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Pair of Lines...

Pair of Lines Passing Though Origin

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Q.

$a x^{2} + 2 h x y + b y^{2} = 0$ always represents a pair of straight lines passing through the origin. If
$\begin{matrix} Column 1 & Column 2 a. h^{2} > a b & 1. Lines are coincident b. h^{2} = a b & 2. Lines are real and distinct c. h^{2} < a b 3. Lines are imaginary with real point of intersection i.e. (0, 0) \end{matrix}$

a-1, b-2, c-3
a-2, b-1, c-3
a-3, b-2, c-1
a-3, b-1, c-2

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Q. If one of the lines given by the equation

2 x^{2} + p x y + 3 y^{2} = 0,

coincides with one of those given by

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

and the other lines represented by them are perpendicular, then

(p, q)

can be

$(- 5, 1)$
$(5, 1)$
$(5, - 1)$
$(- 5, - 1)$

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Q. If

a x^{3} + b y^{3} + c x^{2} y + d x y^{2} = 0

represents three distinct straight lines such that each line bisects the angle between the other two, then which of the following is (are) CORRECT?

$3 b + c = 0$
$3 c + b = 0$
$d^{2} > 3 b c$
$d^{2} < 3 b c$

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Q.

The straight lines represented by the equation $a x^{2} + 2 b x y + b y^{2} = 0$ are perpendicular if

a+b = 0
a-b = 0
a+2h+b = 0
a = 2h

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Q.

Find the equation of the lines joining the origin to the points of intersection of the straight line y=3x+2 with the curve $x^{2} + 2 x y + 3 y^{2} + 4 x + 8 y - 11 = 0$

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

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Q. If the pairs of lines

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

and

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

have exactly one line in common, then the combined equation of the other two lines is given by

$3 x^{2} + 8 x y - 3 y^{2} = 0$
$3 x^{2} + 10 x y + 3 y^{2} = 0$
$3 x^{2} - 2 x y - y^{2} = 0$
$x^{2} + 2 x y - 3 y^{2} = 0$

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Q.

Equation $6 x^{2} - 5 x y + y^{2} = 0$ represents____________________

Pair of coincident lines
Pair of imaginary lines with real point of intersection i.e.(0, 0)
None of these
Pair of distinct straight lines

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Q. Let

L_{1} : 2 x + 3 y = 5

and

L_{2}

be equal sides of a triangle. If

L : x + y = 2

is the perpendicular bisector of the third side, then the equation of

L_{2}

is

$3 x + y = 4$
$x + 5 y = 6$
$x = y$
$3 x + 2 y = 5$

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Q. If the slope of one of the lines given by

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

, be the square of the slope of the other, then

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

is

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Q.

If the lines represented by the equation $2 x^{2} - 3 x y + y^{2} = 0$ make angles $α$ and $β$ with x - axis, then $c o t^{2} α + c o t^{2} β =$

0
3/2
7/4
5/4

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Q. What is the maximum slope of the curve

y = - x^{3} + 3 x^{2} + 2 x - 27 ?

$2$
$5$
$21$
$12$

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Q. If the straight line through the point

P (3, 4)

makes an angle

\frac{π}{6}

with the x-axis and meets the line

3 x + 5 y + 1 = 0

at

Q

, the length

P Q

is

$\frac{132}{12 \sqrt{3} - 5}$
$\frac{132}{5 \sqrt{3} - 12}$
$\frac{132}{5 \sqrt{3} + 12}$
$\frac{132}{12 \sqrt{3} + 5}$

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Q. The angle between the lines

\sqrt{3 x} + y + 1 = 0

and

x + 1 = 0

is.

$\frac{π}{3}$
$\frac{π}{6}$
$\frac{π}{2}$
$\frac{π}{4}$

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Q. If the pairs of lines

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

and

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

have exactly one line in common, then the combined equation of the other two lines is given by

$3 x^{2} + 8 x y - 3 y^{2} = 0$
$3 x^{2} + 10 x y + 3 y^{2} = 0$
$3 x^{2} - 2 x y - y^{2} = 0$
$x^{2} + 2 x y - 3 y^{2} = 0$

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Q. If one of the lines given by the equation

2 x^{2} + p x y + 3 y^{2} = 0,

coincides with one of those given by

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

and the other lines represented by them are perpendicular, then

(p, q)

can be

$(- 5, 1)$
$(5, 1)$
$(5, - 1)$
$(- 5, - 1)$

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Q. If

a x^{3} + b y^{3} + c x^{2} y + d x y^{2} = 0

represents three distinct straight lines such that each line bisects the angle between the other two, then which of the following is (are) CORRECT?

$3 b + c = 0$
$3 c + b = 0$
$d^{2} > 3 b c$
$d^{2} < 3 b c$

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Q.

If the pairs of lines $x^{2} + 2 x y + a y^{2} = 0$ and $a x^{2} + 2 x y + y^{2} = 0$ have exactly one line in common then the joint equation of the other two lines is given by

$3 x^{2} + 8 x y - 3 y^{2} = 0$
$3 x^{2} + 10 x y + 3 y^{2} = 0$
$y^{2} + 2 x y - 3 x^{2} = 0$
$x^{2} + 2 x y - 3 y^{2} = 0$

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Q. If

A_{1}

and

A_{2}

respectively represents the area bounded by the curves

f (x, y)

4 x^{2} \leq y \leq 3 x

and

g

(x, y)

4 x^{2} \leq y \leq | 3 x |

then

A_{1}

:

A_{2}

equals:

2:1
1:2
1:3
3:1

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Q. A unit vector in the

x y

-plane that makes an angle of

45^{0}

with the vector

^i +^j

and an angle of

60^{0}

with the vector

3^i + 4^j

is

$^i$
$\frac{^i +^j}{\sqrt{2}}$
None of these
$\frac{^i -^j}{\sqrt{2}}$

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Q. A tangent having slope of

- \frac{4}{3}

to the ellipse

\frac{x^{2}}{18} + \frac{y^{2}}{32} = 1

intersects the major and minor axes in points

A

and

B

respectively. If

O

is the origin, find the area of

△ O A B

.

$12$
$24$
$48$
$60$

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Q.

Equation $6 x^{2} - 5 x y + y^{2} = 0$ represents____________________

Pair of coincident lines
Pair of imaginary lines with real point of intersection i.e.(0, 0)
Pair of distinct straight lines
None of these

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Q.

A

straight line through the origin

O

meets the parallel lines

4 x + 2 y = 9

and

2 x + y + 6 = 0

at points

P

and

Q

respectively, then point

O

divides the segment

P Q

in the ratio:

$1 : 2$
$4 : 3$
$2 : 1$
$3 : 4$

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Q. Let

f : R \to R

be a positive increasing function with

lim x \to \infty \frac{f (3 x)}{f (x)} = 1

. Then

lim x \to \infty \frac{f (2 x)}{f (x)}

$\frac{3}{2}$
$3$
$1$
$\frac{2}{3}$

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Q.

If the pairs of lines $x^{2} + 2 x y + a y^{2} = 0$ and $a x^{2} + 2 x y + y^{2} = 0$ have exactly one line in common then the joint equation of the other two lines is given by

$3 x^{2} + 8 x y - 3 y^{2} = 0$
$3 x^{2} + 10 x y + 3 y^{2} = 0$
$y^{2} + 2 x y - 3 x^{2} = 0$
$x^{2} + 2 x y - 3 y^{2} = 0$

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Q.

If the lines represented by the equation $2 x^{2} - 3 x y + y^{2} = 0$ make angles $α$ and $β$ with x - axis, then $c o t^{2} α + c o t^{2} β =$

0
$\frac{3}{2}$
$\frac{7}{4}$
$\frac{5}{4}$

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Q. In the symmetrical representation of three or more terms helps to solve sum or product of three terms so that in one of the equations ______

a vanishes.
The equation is no longer valid.
d completely vanishes.
a and d both vanish.

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Q. Find the measure of acute angle between the lines represented by

2 x^{2} + 7 x y + 3 y^{2} = 0

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Q. Find the angle of intersection of the curves:

x^{2} + y^{2} = a^{2} \sqrt{2}

and

x^{2} - y^{2} = a^{2} .

$\frac{π}{4}$
$\frac{π}{2}$
$\frac{π}{3}$
$\frac{π}{6}$

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Q. A circle is described with minor axis of the ellipse as diameter. If the foci lie on the circle, then the eccentricty of the ellipse is

$\frac{1}{\sqrt{3}}$
$\frac{1}{\sqrt{2}}$
$\frac{1}{\sqrt{5}}$
$\frac{1}{2}$

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Q. The slope of the line

| z 1 | = | z + i |

is

$0$
$1 / 2$
$2$
$- 1$

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