Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 52

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The two lines pairs $y^{2} - 4 y + 3 = 0$ and $x^{2} + 4 x y + 4 y^{2} - 5 x - 10 y + 4 = 0$ enclose a $4$ sided convex polygon, then the correct statement among the following is/are

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Important Questions on Point and Straight Line

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 53

If the distance between the lines represented

9 x^{2} - 24 x y + 16 y^{2} + k (6 x - 8 y) = 0

is

4,

then

k

may be

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-2>Q 54

Coordinates of the vertices

B

and

C

of a triangle

A B C

are

(2, 0)

and

(8, 0)

, respectively. The vertex

A

is varying in such a way that

4 \tan \frac{B}{2} \cdot \tan \frac{C}{2} = 1 .

Then locus of

A

is

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 1

The number of complex numbers

z

such that

| z - 1 | = | z + 1 | = | z - i |

equals

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 2

Let

a, λ, μ \in R

. Consider the system of linear equations

a x + 2 y = λ

and

3 x - 2 y = μ

Which of the following statement(s) is/are correct?

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 3

Let

O (0, 0), P (3, 4), Q (6, 0)

be the vertices of a triangle

O P Q .

The point

R

inside the triangle

O P Q

is such that the triangles

O P R, P Q R, O Q R

are of equal area. The coordinates of

R

are

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 4

Lines $L_{1} : y - x = 0$ and $L_{2} : 2 x + y = 0$ intersect the line $L_{3} : y + 2 = 0$ at $P$ and $Q$ respectively. The bisector of the acute angle between $L_{1}$ and $L_{2}$ intersects $L_{3}$ at $R$ .

Statement $I :$ The ratio $P R : R Q$ equals $2 \sqrt{2} : \sqrt{5}$ .

Statement $II :$ In any triangle, bisector of an angle divides the triangle into two similar triangles.

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 5

Consider three points

P = \{- \sin (β - α), - \cos β\}, Q = \{\cos (β - α), \sin β\}

and

R = \{\cos (β - α + θ), \sin (β - θ)\}

, where

0 < α, β, θ < \frac{π}{4}

, then

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Alpha Question Bank for Engineering: Mathematics>Chapter 15 - Point and Straight Line>Exercise-3>Q 6

The locus of the orthocentre of the triangle formed by the

lines (1 + p) x - p y + p (1 + p) = 0,

(1 + q) x - q y + q (1 + q) = 0

and

y = 0,

where

p \neq q,

is