Question

The angle between the lines x + 2y = 11 and 2x – y = 9 is​

Answer 1

Given :- The angle between the lines x + 2y = 11 and 2x – y = 9 is ?

Solution :-

writing both given lines in the form of y = mx + c ,

→ x + 2y = 11

→ 2y = - x + 11

→ y = (-1/2)x + (11/2) => mx + c

→ m = (-1/2)

and,

→ 2x - y = 9

→ y = 2x - 9 => mx + c

→ m = 2 .

So,

• m1 = (-1/2)
• m2 = 2 .

Now, we know that,

• Acute angle between two lines whose slope are m1 and m2 is = | (m1 - m2) / (1 + m1 * m2) |

then,

→ tan A = | {(-1/2) - 2)} / {1 + (-1/2) * 2} |

→ tan A = | (-5/2) / {1 + (-1)} |

→ tan A = | (-5/2) / 0 |

→ tan A = undefined

→ tan A = tan 90°

→ A = 90° .

Hence, the angle between given lines is 90° .

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Answer 2

SOLUTION :-

TO DETERMINE :-

The angle between the lines x + 2y = 11 and 2x – y = 9

CONCEPT TO BE IMPLEMENTED :-

The general equation of any line is

ax + by + c = 0

Which can be rewritten in the Slope - intercept form as

y = mx + c

Where m is the slope of the line

PROCEDURE FOR CHECKING PERPENDICULAR Lines :-

STEP : 1

First write the equation of the two lines

STEP : 2

Rewrite the given equations in Slope - Intercept form

STEP : 3

Find the slope of the two lines

$\sf{Let \: they \: are \: \: m_1 \: \: and \: \: m_2}$

STEP : 4

Find the product of two two slopes

If the product = - 1 i.e

$\sf{m_1 \: \times \: m_2 = - 1}$

Then the given two lines are perpendicular

EVALUATION :-

Here the given equation of the lines are

$\sf{x + 2y = 11} \: \: \: ......(1)$

$\sf{2x - y = 9} \: \: \: \: ....(2)$

Equation (1) can be rewritten as

$\displaystyle \sf{ y = - \frac{x}{2} - \frac{11}{2} \: }$

$\displaystyle \sf{ \therefore \: Slope \: of \: first \: line \: = m_1 = - \frac{1}{2} }$

Equation (2) can be rewritten as

$\sf{y = 2x - 9}$

$\sf{ \therefore \: Slope \: of \: the \: second \: line \: = m_2 = 2}$

Now

$\sf{m_1 \: \times \: m_2 }$

$\displaystyle \sf{ = - \frac{1}{2} \times 2 }$

$\sf{ = - 1}$

Hence the given two lines are perpendicular

Therefore the angle between given two lines = 90°

FINAL ANSWER

The angle between the lines x + 2y = 11 and 2x – y = 9 is 90°

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