Question

What is the locus of the midpoint of chords of ($x-1{)}^2+y^2=1$ that passes through the origin.

• A. x² + y² - x = 0
• B. x² + y² - 3x = 0
• C. x² + y² - y = 0
• D. x² + y² - 3y = 0

Answer 1

The correct option is A

x² + y² - x = 0

Let's draw the given circle first. Circle is centered at (1, 0) and has radius equal to 1 unit.

Here it would be helpful if we could use the formula for the chord centered at a given point.

Since here we want to find the locus of that midpoint, lets take that point to be (h, k)

The formula is,

$T_1=S_1$

Given circle is,

$(x-1{)}^2+y^2=1$

i.e., $x^2-2x+1+y^2=1$

i.e., $x^2+y^2-2x=0$

using this in $T_1=S_1$

$T_1=S_1$

i.e., $hx+yk-(x+h)=h^2+k^2-2h.$

given that this line passes through origin. So we can put $x=0$ and $y=0$ for the above equation.

$0+0-h=h^2+k^2-2h$

i.e., $h^2+k^2-h=0$

$\Rightarrow x^2+y^2-x=0$ is the required locus of the midpoints.