Question

Find the equation of the following parabolas :-
a. Directrix x = 0 , focus at (6,0)
b. Focus at (2,3) , directrix
x - 4y + 3 = 0. Also, find the equation of the axis.
c. Focus (a,b) directrix
$\dfrac{x}{y} + \dfrac{y}{b} = 1$
Help me ​

Answer 1

Answer:

I

Step-by-step explanation:

Answer 2

★ Concept

For part (a)

let P (x,y) be any point on a parabola. Then, by definition of parabola, distance of P from the focus = its distance from the directrix

$∴ \rm \sqrt{(x - 6) ^{2} + (y - 0) ^{2} } = |x|$

⇒ x² - 12x + 36 + y² = x²

⇒ y² = 12x - 36, which is the required equation.

For part (b)

let P (x,y) be any point on a parabola.By definition, the distance of P from (2,3) is equal to the perpendicular distance of P from equation x - 4y + 3 = 0

Therefore,

$⇒ \rm \sqrt{(x - 2) ^{2} + (y - 3) ^{2} }$

$\rm ⇒ \dfrac{ |x - 4y + 3| }{ \sqrt{17} }$

$\sf \: ⇒(x - 2) ^{2} + (y - 3)^{2}$

$\sf ⇒ \dfrac{(x - 4y + 3) ^{2} }{17}$

$\sf \: ⇒ 17(x² - 4x + 4 + y² - 6y + 9)$

$\sf \: ⇒ x² + 16y² + 9 - 8xy + 6x - 24y ,$

16x² + 8xy + y² - 74x - 78y + 212 = 0, which is the required equation.

The axis of the parabola is the line passing through the focus (2,3) and perpendicular to the directrix as given

⇒ x - 4y + 3 = 0

Equation of any line perp. to the line x - 4y + 3 = 0 is 4x + y + k = 0, where k is any arbitrary constant. If it passes through the point (2,3), then

⇒ 4 × 2 + 3 + K = 0

⇒ K = -11

Hence, from (a) the equation of axis is 4x + y - 11 = 0.

Note :

It can be easily verified that the second degree terms 16x² + 8xy + y² form a perfect square, namely, (4x+y)²

For part (c)

let P(x,y) be any point on the parabola. Then,

Distance of P from the focus = √(x-a)² + (y-b)²

Distance of P from the directrix

$\sf \dfrac{x}{y} + \dfrac{y}{b} = 1 \qquad \: bx + ay - ab = 0$

$\sf⇒ | \dfrac{bx + ay - ab}{ \sqrt{ {a}^{2} + {b}^{2} } } |$

By the definition of parabola

distance of P from the focus = its distance from directrix

$\sf \sqrt{ (x-a)² + (y-b)²} = | \dfrac{bx + ay - ab}{ \sqrt{ {a}^{2} + {b}^{2} } } |$

$\sf ⇒ \: (x - a)^{2} + (y - b)^{2} = \dfrac{(bx + ay - ab)^{2} }{ {a}^{2} + {b}^{2} }$

$\sf ⇒ (a² + b²)( x² + y² - 2ax - 2by + a² + b²)$

$\sf⇒ b²x² + a²y² + a²b² + 2abxy - 2a²by - 2ab²x$

$\sf ⇒ \red{\bf a²x² - 2abxy + b²y² - 2a³x - 2b³y + (a⁴ + a²b² + b⁴) = 0}$

which is the required equation of the parabola.

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