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If the points \( \Large \left(a_{1},b_{1}\right), \left(a_{2},b_{2}\right) \) and \( \Large \left( a_{3},b_{3} \right) \) are collinear then lines \( \Large a_{i}x+b_{i}y+1=0 \) for i = 1, 2, 3 are:

A) concurrent
B) indentical
C) parallel
D) none of these

Correct Answer:

A) concurrent

Description for Correct answer:

For collinearity\( \begin{vmatrix}
a_{1} & b_{1} & 1 \\
a_{2} & b_{2} & 1 \\
a_{3} & b_{3} & 1
\end{vmatrix} = 0 \)

For concurrency to lines a \( \Large a_{1}x+b_{1}y+1=0,\ i=1,\ 2,\ 3 \) we have,

\( \begin{vmatrix}
a_{1} & b_{1} & 1 \\
a_{2} & b_{2} & 1 \\
a_{3} & b_{3} & 1
\end{vmatrix} = 0 \) so lines are concurrent.

Part of solved Rectangular and Cartesian products questions and answers : >> Elementary Mathematics >> Rectangular and Cartesian products

Comments

Similar Questions

1). A point P\( \Large \left(2,\ 4\right) \) translates to the point Q along the parallel to the positive direction of x-axis by 2 unit. If O be the origin, then \( \Large \angle OPQ \) is:

A). \( \Large \sin^{-1}\sqrt{\frac{399}{400}} \)

B). \( \Large \cos^{-1} \left(\frac{1}{20}\right) \)

C). \( \Large -\sin^{-1} \left(\sqrt{\frac{399}{400}}\right) \)

D). none of these

2). The line joining \( \Large A \left(b \cos \alpha ,\ b \sin \alpha \right)\ and\ B \left(a \cos \beta ,\ a \sin \beta \right) \) is produced to the point \( \Large M \left(x,\ y\right) \) so that \( \Large AM:MB=b:a\ then\ x \cos \left(\frac{ \alpha + \beta }{2}\right)\ +\ y \sin \left(\frac{ \alpha + \beta }{2}\right) \) is:

A). -1

B). 0

C). 1

D). \( \Large a^{2} + b^{2} \)

3). If \( \Large P_{1},\ P_{2} \) denote the length of perpendiculars from the origin on the lines \( \Large x \sec \alpha + y \cosh \alpha = 2 \alpha \ and\ x \cos \alpha + y \sin \alpha = \alpha \cos 2 \alpha \) respectively, then \( \Large \left(\frac{P_{1}}{P_{2}}+\frac{P_{2}}{P_{1}}\right)^{2} \) is equal to:

A). \( \Large 4 \sin^{2} 4 \alpha \)

B). \( \Large 4 \cos^{2} 4 \alpha \)

C). \( \Large 4 cosec^{2} 4 \alpha \)

D). \( \Large 4 \sec^{2} 4 \alpha \)

4). lf every point on the line \( \Large \left(a_{1}-a_{2}\right)x+ \left(b_{1}-b_{2}\right)y=c \) is equidistant from the points \( \Large \left(a_{1},\ b_{1}\right) \) then 2c is equal to:

A). \( \Large a^{2}_{1}+b^{2}_{1}+a^{2}_{2}-b^{2}_{2} \)

B). \( \Large a^{2}_{1}+b^{2}_{1}+a^{2}_{2}+b^{2}_{2} \)

C). \( \Large a^{2}_{1}+b^{2}_{1}-a^{2}_{2}-b^{2}_{2} \)

D). none of these

5). If two vertices of a triangle are \( \Large \left(-2,\ 3\right)\ and\ \left(5,\ -1\right) \) orthocentre lies at origin and centroid on the line x + y = 7, then the third vertex lies at:

A). \( \Large \left(7,\ 4\right) \)

B). \( \Large \left(8,\ 14\right) \)

C). \( \Large \left(12,\ 21\right) \)

D). none of these

6). The locus of a points p which moves such that 2PA = 3PB, where \( \Large A \left(0,\ 0\right)\ and\ B \left(4,\ -3\right) \) are points is:

A). \( \Large 5x^{2}-5y^{2}-72x+54y+225=0 \)

B). \( \Large 5x^{2}+5y^{2}-72x+54y+225=0 \)

C). \( \Large 5x^{2}+5y^{2}+72x-54y+225=0 \)

D). \( \Large 5x^{2}+5y^{2}-72x-54y+225=0 \)

7). If \( \Large A \left(-a,0\right) \) and \( \Large B \left(a,0\right) \) are two fixed points, then the locus of the point at which AB subtends a right angle is;

A). \( \Large x^{2}+y^{2}=2a^{2} \)

B). \( \Large x^{2}-y^{2}=a^{2} \)

C). \( \Large x^{2}+y^{2}+a^{2}=0 \)

D). \( \Large x^{2}+y^{2}=a^{2} \)

8). If \( \Large a+b+c=0 \), then \( \Large a^{3}+b^{3}+c^{3} \) is equal to

A). \( \Large a^{2} \left(b+c\right)+b^{2} \left(c+a\right)+c^{2} \left(a+b\right) \)

B). \( \Large 3 \left(b+c\right) \left(c+a\right) \left(a+b\right) \)

C). \( \Large 3abc \)

D). \( \Large 6a^{2}b^{2}c^{2} \)

9). Largest number among \( \Large 2^{2^{2}},\ 2^{22},\ 222,\ \left(22\right)^{2} \) is

A). \( \Large 2^{22} \)

B). \( \Large 2^{2^{2}} \)

C). 222

D). \( \Large \left(22\right)^{2} \)

10). Both addition and multiplication of numbers are operations which are

A). commutative but not associative

B). commutative and associative

C). associative but not commutative

D). neither commutative nor associative