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प्रश्न
A line AB meets the x-axis at point A and y-axis at point B. The point P(−4, −2) divides the line segment AB internally such that AP : PB = 1 : 2. Find:
- the co-ordinates of A and B.
- equation of line through P and perpendicular to AB.
उत्तर
∵ Line AB intersects x-axis at A and y-axis at B.
i. Let co-ordinates of A be (x, 0) and of B be (0, y)
Point P(−4, –2) intersects AB in the ratio 1 : 2 internally
∴ `-4 = (1 xx 0 + 2 xx x)/(1 + 2)` ...`(x = (m_1x_2 + m_2x_1)/(m_1 + m_2))`
`\implies -4 = (0 + 2x)/3`
`\implies` 2x = –12
∴ `x = (-12)/2 = -6` and `-2 = (1 xx y + 2 xx 0)/(1 + 2)` ...`[∵ y = (m_1y_2 + m_2y_1)/(m_1 + m_2)]`
`\implies - 2 = (y + 0)/3`
`\implies` y = −6
∴ Co-ordinates of A will be (−6, 0) and Co-ordinates of B will be (0, −6)
ii. Now, slope of AB (m1)
= `(y_2 - y_1)/(x_2 - x_2)`
= `(-6 - 0)/(0 - (-6))`
= `(-6)/6`
= −1
∴ Slope of the line perpendicular to AB (m2) = 1 ...(∵ m1m2 = −1)
∴ Equation of line perpendicular to AB and drawn through P(−4, −2) will be
y − y1 = m(x − x1)
`\implies` y + 2 = 1(x + 4)
`\implies` y + 2 = x + 4
`\implies` y = x + 4 − 2
`\implies` y = x + 2
Hence, required equation of the line is y = x + 2
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