Question

Find the equation to the parabola
whose focus and directive are (2, 3),
x - 2y - 6=0​

Answer 1

EXPLANATION.

Equation to the parabola whose focus = (2,3) and directrix = x - 2y - 6 = 0.

According, to the definition of parabola,

⇒ (SP) = (PM).

⇒ (x - 2) + (y - 3) = | x - 2y - 6/√(1)² + (-2)²|.

⇒ (x - 2) + (y - 3) = | x - 2y - 6/√5|.

Squaring on both sides, we get.

⇒ [(x - 2)² + (y - 3)²]² = | x - 2y - 6/√5 |².

⇒ 5[(x - 2)² + (y - 3)²] = x² + 4y² + 36 - 4xy - 12x + 24y.

⇒ 5[x² + 4 - 4x + y² + 9 - 6y] = x² + 4y² + 36 - 4xy - 12x + 24y.

⇒ 5x² + 5y² - 20x - 30y + 65 = x² + 4y² + 36 - 4xy - 12x + 24y.

⇒ 5x² - x² + 5y² - 4y² - 20x + 12x - 30y - 24y + 4xy + 65 - 36 = 0.

⇒ 4x² + y² + 4xy - 8x - 54y + 29 = 0.

                                                                                             

MORE INFORMATION.

Conditions of Normal.

The line y = mx + c is a normal to the parabola,

(1) = y² = 4ax if c = - 2am - am³.

(2) = x² = 4ay if c = 2a + a/m².

Normal Chord.

If the normal at 't₁' meets the parabola y² = 4ax again at the point 't₂' then this is called as normal chord. again for normal chord, t₂ = -t₁ - 2/t₁.

Answer 2

$\large\underline\blue{\bold{Given \: Question :- }}$

• ☆Find the equation to the parabola whose focus is (2,3) and directrix is x - 2y - 6 = 0.

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$\huge \green{AηsωeR} ✍$

$\large\underline\pink{\bold{Given :- }}$

• Focus of parabola is (2,3)
• Equation of directrix = x - 2y - 6 = 0

$\large\underline\purple{\bold{To \: Find :- }}$

• ☆ Equation of Parabola

$\large\underline\red{\bold{Solution:- }}$

☆Let P(x, y) be any point on the parabola having focus (2, 3) and equation of directrix is x - 2y - 6 = 0.

☆Draw PM perpendicular from P on the directrix, then

$\large\purple{\bold{By \: definition \: of \: Parabola }}$

$\blue{\bold{\bf\implies \:SP = PM}}$

$\sf \: \sqrt{ {(x - 2)}^{2} + {(y - 3)}^{2}} = |\dfrac{x - 2y - 6}{ \sqrt{ {(1)}^{2} + {( - 2)}^{2} } } |$

$\blue{\bold{squaring \: both \: sides \: we \: get}}$

$\sf \: ⟼ {(x - 2)}^{2} + {(y - 3)}^{2} = {( |\dfrac{x - 2y - 6}{ \sqrt{5} } | )}^{2}$

$\sf \: ⟼ 5({x}^{2} + 4 - 4x + {y}^{2} + 9 - 6y )= \\ {x}^{2} + {4y}^{2} + 36 - 4xy + 24y - 12x$

$\sf \: ⟼5 {x}^{2} + 65 - 20x + {5y}^{2} - 30y = \\ {x}^{2} + {4y}^{2} + 36 - 4xy + 24y - 12x$

$\sf \: ⟼4 {x}^{2} + {y}^{2} + 4xy - 8x - 54y + 29 = 0$

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