First slide

Equation of line in Straight lines

Question

The line joining $A (b \cos a, b \sin a)$ and $B (a \cos b, a \sin b)$ is produced to the point $M (x, y)$ so that $A M : M B = b : a,$ , then $x \cos \frac{α + β}{2} + y \sin \frac{α + β}{2} =$

Moderate

A

-1
B

0
C

1
D

$a^{2} + b^{2}$

Solution

As $M$ divides $A B$ externally in the ratio $b : a$
$x = \frac{b (a \cos β) - a (b \cos α)}{b - a}$ and
$\begin{array}{l} y = \frac{b (a \sin β) - a (b \sin α)}{b - a} \\ \Rightarrow \frac{x}{y} = \frac{\cos β - \cos α}{\sin β - \sin α} = \frac{2 \sin \frac{α + β}{2} \sin \frac{α - β}{2}}{2 \cos \frac{α + β}{2} \sin \frac{β - α}{2}} \\ \Rightarrow x \cos \frac{α + β}{2} + y \sin \frac{α + β}{2} = 0 \end{array}$

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