Question

Parabola
x² + 2xy +y²+ 4
$4 \sqrt{2x}$
-
$4 \sqrt{2y}$
=0
find relevant points of conuc sections
​

Answer 1

Answer:

Given the focus and the directrix of a parabola, we can find the parabola's equation. Consider, for example, the parabola whose focus is at (-2,5)(−2,5)left parenthesis, minus, 2, comma, 5, right parenthesis and directrix is y=3y=3y, equals, 3. We start by assuming a general point on the parabola (x,y)(x,y)left parenthesis, x, comma, y, right parenthesis.

Using the distance formula, we find that the distance between (x,y)(x,y)left parenthesis, x, comma, y, right parenthesis and the focus (-2,5)(−2,5)left parenthesis, minus, 2, comma, 5, right parenthesis is \sqrt{(x+2)^2+(y-5)^2}

(x+2)

2

+(y−5)

2

square root of, left parenthesis, x, plus, 2, right parenthesis, squared, plus, left parenthesis, y, minus, 5, right parenthesis, squared, end square root, and the distance between (x,y)(x,y)left parenthesis, x, comma, y, right parenthesis and the directrix y=3y=3y, equals, 3 is \sqrt{(y-3)^2}

(y−3)

2

square root of, left parenthesis, y, minus, 3, right parenthesis, squared, end square root. On the parabola, these distances are equal:

\begin{aligned} \sqrt{(y-3)^2} &= \sqrt{(x+2)^2+(y-5)^2} \\\\ (y-3)^2 &= (x+2)^2+(y-5)^2 \\\\ \blueD{y^2}-6y\goldD{+9} &= (x+2)^2\blueD{+y^2}\maroonD{-10y}+25 \\\\ -6y\maroonC{+10y}&=(x+2)^2+25\goldD{-9} \\\\ 4y&=(x+2)^2+16 \\\\ y&=\dfrac{(x+2)^2}{4}+4\end{aligned}

(y−3)

2

(y−3)

2

y

2

−6y+9

−6y+10y

4y

y

=

(x+2)

2

+(y−5)

2

=(x+2)

2

+(y−5)

2

=(x+2)

2

+y

2

−10y+25

=(x+2)

2

+25−9

=(x+2)

2

+16

=

4

(x+2)

2

+4

Step-by-step explanation:

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