Question

The equation of the image of pair of lines $y = \left| {x - 1} \right|$ with respect to $y$ axis is
A.${x^2} - {y^2} - 2x + 1 = 0$
B.${x^2} - {y^2} - 4x + 4 = 0$
C.$4{x^2} - 4x - {y^2} + 1 = 0$
D.${x^2} - {y^2} + 2x + 1 = 0$

Answer 1

Hint: First of all, remove the modulus sign by squaring both sides. For, finding the image of lines $y = \left| {x - 1} \right|$ with respect to $y$ axis, substitute $x$as $- x$ in the resultant equation. Simplify the result to get the required answer.

Complete step by step answer:

We are given the equation of pair of lines as $y = \left| {x - 1} \right|$
We will first remove the mode by squaring both sides of the equation.
As we know, ${\left( {\left| a \right|} \right)^2} = {a^2}$
On squaring both sides, we get,
${y^2} = {\left( {\left| {x - 1} \right|} \right)^2} \\\ {y^2} = {\left( {x - 1} \right)^2}{\text{ }}\left( 1 \right) \\\$
We have to find the image of pair of lines $y = \left| {x - 1} \right|$ with respect to $y$ axis.
We can find the image by substituting the value of $x$as $- x$ in the equation (1).
Therefore, we get, ${y^2} = {\left( { - x - 1} \right)^2} = {\left( {x + 1} \right)^2}$
Simplify the above equation using the formula, ${\left( {a + b} \right)^2} = {a^2} + {b^2} + 2ab$
On simplifying, we will get,
${y^2} = {x^2} + 1 + 2x$
On rearranging the equation we will get,
${x^2} - {y^2} + 2x + 1 = 0$
Hence, option D is correct.

Note: If we want to find the image with respect to $y$ axis, then we will substitute the value of $x$as $- x$and if we want to find the image with respect to $x$ axis, then we will substitute the value of $y$as $- y$ in the formed equation or given equation.