Answer:

x² + y² - 14x + 2y + 2xy + 17 = 0

4x² + y² - 4XY + 104x + 148y - 124 = 0

Step-by-step explanation:

(1)

Given Equation is x = y - 1

It can be written as, x - y = -1     ---- (i)

-y = -x - 1

y = x + 1

∴ Slope = 1.

∴ Slope of axis = -1.

Equation of axis is y - 1 = -1(x - 2)

y - 1 = -x + 2

y - 1 + x - 2 = 0

x + y - 3 = 0     ---- (ii)

On solving (i) & (ii), we get

x - y = -1

x + y = 3

------------

2X = 2

x = 1

Substitute x = 1 in (i), we get

x - y = -1

1 - y = -1

y = 2

∴ The directrix and axis intersects at A(1,2)

LET h(s,k) be the focus of parabola.

∴ V(2,1) is the mid-point of AS.

∴ 2 = 1 + h/2 and 1 = 2 + k/2

h = 3, k = 0.

∴ Focus is S(3,0).

Let P(x,y) be any point on parabola.

PS = PM

⇒√(x - 3)² + (y - 0)² = |x -y + 1/√1 + 1|

On Squaring both sides, we get

⇒ (x - 3)² + y² = (x - y + 1)^2/2)

⇒ x² + 9 - 6y + y² = x² + y² + 1 - 2xy - 2y + 2x/2

⇒ 2x² + 18 - 12y + 2y² = x² + y² + 1 - 2xy - 2y + 2x

⇒ x² + y² - 14x + 2y + 2xy + 17 = 0

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(2)

Let the focus be S(-6,-6) and vertex be A(-2,2)

Let z(x,y) be the projection of S on directrix. A is the mid-point of SZ.

⇒ (-2,2) = (-6 + x/2, -6 + y/2)

⇒ x = 2, y = 10.

∴ Z = (2,10)

Slope of directrix = -1/(-6 - 2/-6 + 2)

                            = -1/2..

Equation of directrix is y - 10 = -1/2(x - 2)

⇒ 2y - 20 = -x + 2

⇒ 2y - 20 + x - 2 = 0

⇒ x + 2y - 22 = 0.

Let P(x,y) be any point on the parabola, then

SP = PM

SP² = PM²{PM is the perpendicular from P to the directrix}

⇒ (x + 6)² + (y + 6)² = (x+ 2y - 22)²/1² + 4

⇒ x² + 36 + 12x + y² + 36 + 12y = x² + 4xy - 44x + 4y² - 88y + 484/5

⇒ 5(x² + 36 + 12x + y² + 36 + 12y) = x² + 4xy - 44x + 4y² - 88y + 484

⇒ 5x² + 60x + 5y² + 60Y + 360 = x² + 4xy - 44x + 4y² - 88y + 484

⇒ 5x² + 104x - 4yx + y² + 148y - 124 - x² = 0

⇒ 4x² + y² - 4xy + 104x + 148y - 124 = 0

(or)

(2x - y)² + 104x + 148y - 124 = 0

Hope it helps!