Question

A straight line through the point $(1,1)$ meets the X-axis at 'A' and the Y-axis at 'B'. The locus of the mid point of AB is

• A. $2xy+x+y=0$
• B. $x+y-2xy=0$
• C. $x+y+2=0$
• D. $x+y-2=0$

Answer 1

Answer: (B)

$x+y-2xy=0$

The equation of the straight line in intercept form is

$\frac{x}{a}+\frac{1}{b}=1$ ..... (i)

Where a & b are shown in the figure

Since it passes through P (1, 1).

$\Rightarrow \frac{1}{a}+\frac{1}{b}=1$ or $a+b=ab$ ....... (ii)

Let the coordinates of the mid-point M of AB are (h, k) $\equiv (\frac{a}{2},\frac{b}{2})$

Substitute the values of a and b in (ii), we get $2h+2k=2h\times 2k\Rightarrow h+k=2hk$

Hence, the equation of the locus become

$x+h=2hk\Rightarrow x+y-2xy=0$