The image of the parabola x² = 4y in the line x − y = 1 is

The correct answer is: (y + 1)² = 4(x − 1)

To find the equation of the image of the parabola, we can find the image of a general point on the parabola and then determine the new equation.

Step 1: Understand the transformation of coordinates
Let $(x,y)$ be any point on the given parabola:
$x^2=4y$
Let $(x^{\prime },y^{\prime })$ be the image of the point $(x,y)$ in the line:
$x-y=1$ which can be written as $x-y-1=0$

Step 2: Use the formula for the image of a point about a line
The image of a point $(x,y)$ with respect to the line $ax+by+c=0$ is given by:
$\frac{x^{\prime }-x}{a}=\frac{y^{\prime }-y}{b}=-2\frac{ax+by+c}{a^2+b^2}$

Here, the line is $1x-1y-1=0$, so we have $a=1$, $b=-1$, and $c=-1$.
Substituting these values:
$\frac{x^{\prime }-x}{1}=\frac{y^{\prime }-y}{-1}=-2\frac{x-y-1}{1^2+(-1{)}^2}$

Simplifying the right-hand side:
$-2\frac{x-y-1}{2}=-(x-y-1)=-x+y+1$

Step 3: Solve for original coordinates in terms of transformed coordinates
From the first equality:
$x^{\prime }-x=-x+y+1$
$\Rightarrow x^{\prime }=y+1$
$\Rightarrow y=x^{\prime }-1$

From the second equality:
$-(y^{\prime }-y)=-x+y+1$
$\Rightarrow y^{\prime }-y=x-y-1$
$\Rightarrow y^{\prime }=x-1$
$\Rightarrow x=y^{\prime }+1$

Step 4: Substitute back to find the new equation
Substitute the expressions for $x$ and $y$ into the original parabola equation $x^2=4y$:
${(y^{\prime }+1)}^2=4(x^{\prime }-1)$

Replacing the current coordinates $(x^{\prime },y^{\prime })$ with general $(x,y)$ variables:
${(y+1)}^2=4(x-1)$