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प्रश्न
M is the mid-point of the line segment joining the points A(0, 4) and B(6, 0). M also divides the line segment OP in the ratio 1 : 3. Find :
- co-ordinates of M
- co-ordinates of P
- length of BP
उत्तर
M is mid point of the line segment joining the points A(0, 4) and B(6, 0) M divides the line segment OP in the ratio 1 : 3
i. Now co-ordinates of M = `((0 + 6)/2, (4 + 0)/2) = (3, 2)`
ii. Let co-ordinates of P be (x, y), O is (0, 0)
`3 = (m_1x_2 + m_2x_1)/(m_1 + m_2)`
`3 = (1 xx x + 3 xx 0)/(1 + 3)`
= `(x + 0)/4`
= `x/4`
`=>` x = 12 and `2 = (1*y + 3 xx 0)/(1 + 3) = (y + 0)/4 = y/4`
∴ y = 2 × 4 = 8
∴ Co-ordinates of P are (12, 8)
iii. Length of BP = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
= `sqrt((2 - 6)^2 + (8 - 0)^2`
= `sqrt(6^2 + 8^2)`
= `sqrt(36 + 64)`
= `sqrt(100)`
= 10 units
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= `a(1/t^2 + 1) = (a(t^2 + 1))/t^2`
Now `(1)/"SP" + (1)/"SQ" = (1)/(a(t^2 + 1)) + (1 xx t^2)/(a(t^2 + 1)`
= `((1 + t^2))/(a(t^2 + 1)`
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