Question

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Answer 1

${\qquad\qquad\huge\underline{{\sf Answer}}}$

Let's solve ~

Equation of directrix is : y = 1, so we can say that it's a parabola of form : -

$\qquad \sf  \dashrightarrow \: (x - h) {}^{2} = 4a(y - k)$

• h = x - coordinate of focus = -4

• k = y - coordinate of focus = 5

• a = half the perpendicular distance between directrix and focus = 1/2(5 - 1) = 1/2(4) = 2

and since the focus is above the directrix, it's a parabola with upward opening.

$\qquad \sf  \dashrightarrow \: (x - ( - 4)) {}^{2} = 4(2)(y - 5)$

$\qquad \sf  \dashrightarrow \: (x + 4) {}^{2} = 8(y - 5)$

$\qquad \sf  \dashrightarrow \: {x}^{2} + 8x + 16 = 8y - 40$

$\qquad \sf  \dashrightarrow \: 8y = {x}^{2} + 8x + 56$

$\qquad \sf  \dashrightarrow \: y = \cfrac{1}{8} {x}^{2} + x + 7$

Answer 2

Directrix

• y=1

Focus

• (h,k)=(-4,5)

Focus lies in Q3 and above y=1

• Parabola is opening upwards

Then

Perpendicular distance

• (5-1)=4

Find a for the equation

• a=4/2=2

Now the equation is

$\\ \rm\dashrightarrow 4a(y-k)=(x-h)^2$

$\\ \rm\dashrightarrow 4(2)(y-5)=(x+4)^2$

$\\ \rm\dashrightarrow 8(y-5)=x^2+8x+16$

$\\ \rm\dashrightarrow 8y-40=x^2+8x+16$

$\\ \rm\dashrightarrow 8y=x^2+8x+16+40$

$\\ \rm\dashrightarrow 8y=x^2+8x+56$

$\\ \rm\dashrightarrow y=\dfrac{x^2}{8}+x+7$