Question

Find the length of the chord made by 3x^2 - y^2=3 and y +x - 5=0
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Answer 1

SOLUTION

TO DETERMINE

The length of the chord made by 3x² - y² = 3 and y + x - 5 = 0

EVALUATION

Here the given equation of the curve is

3x² - y² = 3 - - - -(1)

The given equation of the line is

y + x - 5 = 0 - - - - - - (2)

Form Equation 2 we get

y = 5 - x - - - - (3)

From Equation 1 and Equation 3 we get

$\sf{3 {x}^{2} - {(x - 5)}^{2} = 3 }$

$\sf{ \implies \: 3 {x}^{2} - {x}^{2} + 10x - 25 = 3 }$

$\sf{ \implies \: 2 {x}^{2} + 10x - 28 = 0 }$

$\sf{ \implies \: {x}^{2} + 5x - 14 = 0 }$

$\sf{ \implies \: {x}^{2} + (7 - 2)x - 14 = 0 }$

$\sf{ \implies \: {x}^{2} + 7x - 2x - 14 = 0 }$

$\sf{ \implies \: (x + 7)(x - 2) = 0 }$

x + 7 = 0 gives x = - 7

x - 2 = 0 gives x = 2

For x = - 7 we have y = 12

For x = 2 we have y = 3

So the two points are ( - 7, 12 ) & ( 2,3)

Hence the required length of the chord

$\sf{ = \sqrt{ {(2 + 7)}^{2} +{ (3 - 12) }^{2} } }$

$\sf{ = \sqrt{ {(9)}^{2} +{ (9) }^{2} } }$

$\sf{ = 9 \sqrt{2} }$

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Answer 2

Answer:

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