Let the image of parabola x² = 4y, in the line x y = 1 be (y + a)² = b(x - c), 2 a, b, c∈ N. Then a + b + c is equal to

Image of parabola x^2 = 4y in the line x − y = 1

Q. Let the image of parabola

$$ x^2 = 4y $$ in the line $$ x - y = 1 $$ be $$ (y+a)^2 = b(x-c), $$ where $a, b, c \in \mathbb{N}$. Then $a+b+c$ is equal to

(A) 12

(B) 8

(D) 4

Correct Answer: 6

Explanation

Given parabola:

$$ x^2 = 4y $$

Line of reflection:

$$ x - y = 1 $$

To find the image of the parabola in the line, use the reflection transformation. The reflection of a point $(x,y)$ in the line $x-y=1$ is obtained by:

$$ x' = y + 1,\quad y' = x - 1 $$

Replace $x$ by $y+1$ and $y$ by $x-1$ in the original equation.

Original equation:

$$ x^2 = 4y $$

After substitution:

$$ (y+1)^2 = 4(x-1) $$

Expanding:

$$ y^2 + 2y + 1 = 4x - 4 $$ $$ (y+1)^2 = 4(x-1) $$

Comparing with the given form:

$$ (y+a)^2 = b(x-c) $$

We get:

$$ a = 1,\quad b = 4,\quad c = 1 $$

Therefore,

$$ a+b+c = 1+4+1 = 6 $$

Hence, the correct answer is 6.

Explanation

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