A variable line (x/a)+ (y/b)=1 is such that a + b = 4 .The locus of the midpoint of the portion of the line intercepted between the axes is

Question

A variable line (x/a)+ (y/b)=1 is such that a + b = 4 .The locus of the midpoint of the portion of the line intercepted between the axes is (A) x + y = 4 (B) x + y = 8 (C) x + y = 1 (D) x + y = 2

Tardigrade Admin · Accepted Answer

Let the coordinate of mid point of AB is (x1, y1) . <img class="img-fluid question-image" alt="image" src="https://cdn.tardigrade.in/img/question/mathematics/398f76c9b23deeac33a6a0c70491368f-.png" /> ∴ x1 = (a+0/2) , y1 = (0+b/2) ⇒ a = 2x1 , b = 2y1 Given , a+b=4 ∴ 2x1 + 2y1 = 4 ⇒ x1 + y1 = 2 Hence, the locus of the mid point is x+ y = 2

Q. A variable line $\frac{x}{a} + \frac{y}{b} = 1$ is such that $a + b = 4$ .The locus of the midpoint of the portion of the line intercepted between the axes is

$x + y = 4$

$x + y = 8$

$x + y = 1$

$x + y = 2$

Let the coordinate of mid point of $A B$ is $(x_{1}, y_{1})$ .

$∴ x_{1} = \frac{a + 0}{2}, y_{1} = \frac{0 + b}{2}$
$\Rightarrow a = 2 x_{1}, b = 2 y_{1}$
Given , $a + b = 4$
$∴ 2 x_{1} + 2 y_{1} = 4$
$\Rightarrow x_{1} + y_{1} = 2$
Hence, the locus of the mid point is
$x + y = 2$

Q. A variable line ax​+by​=1 is such that a+b=4 .The locus of the midpoint of the portion of the line intercepted between the axes is

x+y=4

x+y=8

x+y=1

x+y=2