Question

The length of the latus rectum of the parabola
13[(x-3)^2+(y-4)^2 )= (2x-3y+ 5)^2 is
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Answer 1

SOLUTION :

TO DETERMINE

The length of the latus rectum of the parabola

$\sf{ 13 \bigg[ {(x - 3)}^{2} + {(y - 4)}^{2}\bigg] = {(2x - 3y + 5)}^{2} \: }$

EVALUATION

Here the given equation of parabola is

$\sf{ 13 \bigg[ {(x - 3)}^{2} + {(y - 4)}^{2}\bigg] = {(2x - 3y + 5)}^{2} \: }$

So the focus of the parabola is ( 3 , 4 )

Also the equation of the directrix is

$\sf{2x - 3y + 5 = 0 \: }$

Now the distance between focus and directrix is

$\displaystyle\sf{ = \bigg| \frac{(2 \times 3) - (3 \times 4) + 5}{ \sqrt{ {2}^{2} + {3}^{2} } } \bigg| }$

$\displaystyle\sf{ = \bigg| \frac{6 - 12 + 5}{ \sqrt{ 4 + 9 } } \bigg| }$

$\displaystyle\sf{ = \frac{1}{ \sqrt{ 13 } } }$

Now we know that Latus rectum is double of the distance between focus and directrix

Hence the length of the latus rectum

$\displaystyle\sf{ = 2 \times \frac{1}{ \sqrt{ 13 } } } \: \: \: unit$

$\displaystyle\sf{ = \frac{2}{ \sqrt{ 13 } } } \: \: \: unit$

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