First slide

Equation of line in Straight lines

Question

$Given A \equiv (1, 1) and A B is any line through it cutting the x -axis at B . If A C is perpendicular to A B and meets the$
$y -axis in C, then the equation of the locus of midpoint P o f B C i s$

Moderate

A

$x + y = 1$
B

$x + y = 2$
C

$x + y = 2 x y$
D

$2 x + 2 y = 1$

Solution

$The equation of line A B is y - 1 = m (x - 1) . Therefore, the equation of line A C is$
$y - 1 = - \frac{1}{m} (x - 1)$
$\begin{aligned} 2 h = 1 - \frac{1}{m} \\ 2 k = 1 + \frac{1}{m} \end{aligned}$
$Eliminating m, we have locus x + y = 1 .$

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