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

Correct Answer:


B) 0

Description for Correct answer:
Therefore, M divides AB in the ratio b : a (externally)

\( \Large x=\frac{ba \cos \beta - ab \cos \alpha }{b-a}\ and\ y=\frac{ab \sin \beta - ab \sin \alpha }{b-a} \)

\( \Large \frac{x}{y} = \frac{\cos \beta - \cos \alpha }{\sin \beta - \sin \alpha } \)

=> \( \Large \frac{x}{y}=\frac{2\sin \left(\frac{ \alpha + \beta }{2}\right)\sin \left(\frac{ \alpha - \beta }{2}\right) }{2\cos \left(\frac{\alpha + \beta}{2} \right)sin \left(\frac{ \beta - \alpha }{2}\right) } \)

=> \( \Large x \cos \frac{ \alpha + \beta }{2}+y \sin \left(\frac{ \beta - \alpha }{2}\right)=0 \)

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

Comments

Similar Questions

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

2). 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

3). 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

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

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

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

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

8). 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

9). The positive square root of \( \Large \left(x^{2}+2x-1\right)+\frac{1}{x^{2}+2x+1} \) is

A). \( \Large \left(x+1\right)+\frac{1}{ \left(x+1\right) } \)

B). \( \Large \left(x+1\right)-\frac{1}{ \left(x+1\right) } \)

C). \( \Large \left(x+2\right)-\frac{1}{ \left(x+1\right) } \)

D). \( \Large \left(x+2\right)+\frac{1}{ \left(x+1\right) } \)

10). The simplified value of the decimal fraction \( \Large \frac{1.59 \times 1.59-.41 \times .41}{1.59-.41} \)

A). 1

B). 1.4

C). 2

D). 2.6