Question

Prove that the equation ?x2+ 2xy + y2-2x-1=0 represents
a parabola and find its focus, latus rectum and directrix.
Also trace the curve.
(10)​

Answer 1

Answer:

I DONT KNOW . TRY YOURSELF .

Answer 2

To prove that the equation x2+ 2xy + y2-2x-1=0 represents a parabola, we can try to simplify it by completing the square.

Starting with the given equation:

x²2 + 2xy + y²2 - 2x - 1 = 0

Rearranging terms:

x²2 - 2x + y²2 + 2xy = 1

Completing the square for the x-terms:

(x - 1)²2 - 1 + y²2 + 2xy = 1

(x - 1)²2 + y²2 + 2xy = 2

Completing the square for the y-term:

(x - 1)²2 + (y + x)²2 = 2

Dividing both sides by 2:

[(x - 1)²2 + (y + x)²2]/2 = 1

This is the equation of a parabola centered at (1, -1/2), with the focus at (1, -3/4) and the directrix at y = -1/4.

To find the latus rectum, we can use the formula l = 2F where F is the distance from the vertex to the focus. The vertex is (1, -1/2), and we know the focus is (1, -3/4), so:

F = (1/4)

l = 2F = 1/2

To graph the parabola, we can plot the vertex, focus, and directrix, and use the fact that the parabola is symmetric to sketch its shape.

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