Question

The equation of the parabola whose focus is the point (2, 3) and directrix is the line x - 4y + 3 = 0 is​

Answer 1

$\large\sf\underline{Given:}$

• Focus ( S ) = ( 2, 3 ) and

• Directrix ( M ) , x – 4y + 3 = 0.

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$\large\sf\underline{To\:find:}$

• Equation of the parabola

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$\large\sf\underline{Solution:}$

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Let us assume P (x, y) be any point on the parabola .

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The distance between two points ( x₁ , y₁ ) and ( x₂ , y₂ ) is given as :

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$\sf\:\sqrt{(x₁- x₂)^{2} + {(y₁- y₂)}^{2} }$

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And the perpendicular distance from the point (x₁ , y₁) to the line ax + by + c = 0 is

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$\sf➞\:\frac{ |ay₁+ by₁ + c| }{ \sqrt{ {a}^{2} + {b}^{2} } }$

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So by equating both, we get

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$\sf➞\:{(x - 2)}^{2} + {(y - 3)}^{2} ={( \frac{ |x - 4y + 3| }{ \sqrt{ {1}^{2} + {( - 4)}^{2} } } })^{2}$

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$\sf➞\:x^{2}-4x+4+y^{2}-6y+9= \frac{ {( |x - 4y + 3|) }^{2} }{1 + 16}$

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$\sf➞\:x^{2}+y^{2}-4x-6y+13= \frac{( {x}^{2} + 16 {y}^{2} + 9 + 6x - 24y - 8xy) }{17}$

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Upon cross multiplication, we get

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$\sf➞\:17 {x}^{2} + 17 {y}^{2} - 68x - 102y + 221={x}^{2} + 16 {y}^{2} + 6x - 24y - 8xy + 9$

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$\sf➞\:16 {x}^{2} + {y}^{2} + 8xy - 74x - 78y + 212 = 0$

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∴ The equation of the parabola is :

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${\sf{{\purple{16 {x}^{2} + {y}^{2} + 8xy - 74x - 78y + 212 = 0}}}}$

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