Question

Describe, sketch and label the focus, vertex and directrix of the parabola

16

ଶ + 24 − 32 + 25 = 0.
​

Answer 1

Answer:

24_________32--------

Answer 2

SOLUTION

TO DETERMINE

Describe, sketch and label the focus, vertex and directrix of the parabola

$\sf{16 {x}^{2} + 24x - 32y + 25 = 0 }$

EVALUATION

Here the given equation of the parabola is

$\sf{16 {x}^{2} + 24x - 32y + 25 = 0 }$

$\displaystyle\sf{ \implies {(4x)}^{2} + 2 \times 4x \times 3 + {(3)}^{2} - 32y + 16 = 0 }$

$\displaystyle\sf{ \implies {(4x + 3)}^{2} = 32y - 16 }$

$\displaystyle\sf{ \implies { \bigg(x + \frac{3}{4} \bigg)}^{2} = 2 \bigg(y - \frac{1}{2} \bigg) }$

Comparing with

$\displaystyle\sf{ \implies { \bigg(x - \alpha \bigg)}^{2} = 4a \bigg(y - \beta \bigg) }$

We get

$\displaystyle\sf{ \implies \alpha = - \frac{3}{4} \: , \: a = \frac{1}{2} \: ,\: \beta = \frac{1}{2} }$

$\displaystyle\sf{ focus = ( \alpha \: , \: a + \beta ) = \bigg( - \frac{3}{4} ,1 \bigg) }$

$\displaystyle\sf{ vertex = ( \alpha \: , \: \beta ) = \bigg( - \frac{3}{4} \: , \: \frac{1}{2} \bigg) }$

Equation of the directrix

$\sf{y + a = \beta }$

$\displaystyle\sf{ \implies \: y + \frac{1}{2} = \frac{1}{2} }$

$\displaystyle\sf{ \implies \: y = 0 }$

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