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Question
Equations of the lines through the point (3, 2) and making an angle of 45° with the line x – 2y = 3 are ______.
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Solution
Equations of the lines through the point (3, 2) and making an angle of 45° with the line x – 2y = 3 are 3x – y – 7 = 0 and x + 3y – 9 = 0.
Explanation:
Given line is x – 2y = 3 and the point is (3, 2)
Equation of a line passing through the point (3, 2) is y – 2 = m(x – 3) ......(i)
Angle between equation (i) and the given line x – 2y = 3
Whose slope is `1/2`
∴ tan θ = `|(m_1 - m_2)/(1 + m_1m_2)|`
⇒ tan 45° = `|(m - 1/2)/(1 + m xx 1/2)|`
⇒ 1 = `|(m - 1/2)/(1 + m/2)|`
⇒ `(m - 1/2)/(1 + m/2) = +- 1`
Taking (+) sign,
`(m - 1/2)/(1 + m/2)` = 1
⇒ `m - 1/2 = 1 + m/2`
⇒ `m - m/2 = 1 + 1/2`
⇒ `m/2 = 3/2`
⇒ m = 3
Taking (–) sign,
`(m - 1/2)/(1 + m/2)` = – 1
⇒ `m - 1/2 = - 1 - m/2`
⇒ `m + m/2 = - 1 + 1/2`
⇒ m = `- 1/3`
So, the required equations are,
When m = 3,
y – 2 = 3(x – 3)
⇒ y – 2 = 3x – 9
⇒ 3x – y – 7 = 0
When m = `- 1/3`
y – 2 = `- 1/3 (x - 3)`
⇒ 3y – 6 = – x + 3
⇒ x + 3y – 9 = 0
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