Advertisements
Advertisements
Question
Find the ratio in which the line 2x + 3y – 5 = 0 divides the line segment joining the points (8, –9) and (2, 1). Also find the coordinates of the point of division.
Solution
Let the line 2x + 3y – 5 = 0 divides the line segment joining the points A(8, –9) and B(2, 1) in the ratio λ : 1 at point P.
∴ Coordinates of P = `{(2λ + 8)/(λ + 1), (λ - 9)/(λ + 1)}` ...`[∵ "Internal division" = {(m_1x_2 + m_2x_1)/(m_1 + m_2), (m_1y_2 + m_2y_1)/(m_1 + m_2)}]`
But P lies on 2x + 3y – 5 = 0
∴ `2((2λ + 8)/(λ + 1)) + 3((λ - 9)/(λ + 1)) - 5` = 0
⇒ 2(2λ + 8) + 3(λ – 9) – 5(λ + 1) = 0
⇒ 4λ + 16 + 3λ – 27 – 5λ – 5 = 0
⇒ 2λ – 16 = 0
⇒ λ = 8
⇒ λ : 1 = 8 : 1
So, the point P divides the line in the ratio 8 : 1.
∴ Point of division P = `{(2(8) + 8)/(8 + 1), (8 - 9)/(8 + 1)}`
= `((16 + 8)/9, - 1/9)`
= `(24/9, (-1)/9)`
= `(8/3, (-1)/9)`
Hence, the required point of division is `(8/3, (-1)/9)`.
APPEARS IN
RELATED QUESTIONS
Find the coordinates of the points which divide the line segment joining A (−2, 2) and B (2, 8) into four equal parts.
In what ratio is the join of (4, 3) and (2, –6) divided by the x-axis? Also, find the co-ordinates of the point of intersection.
The line segment joining the points M(5, 7) and N(–3, 2) is intersected by the y-axis at point L. Write down the abscissa of L. Hence, find the ratio in which L divides MN. Also, find the co-ordinates of L.
A(20, 0) and B(10, –20) are two fixed points. Find the co-ordinates of the point P in AB such that : 3PB = AB. Also, find the co-ordinates of some other point Q in AB such that : AB = 6 AQ.
Find the lengths of the medians of a ΔABC whose vertices are A(0,-1) , B(2,1) and C (0.3).
M and N are two points on the X axis and Y axis respectively. P (3, 2) divides the line segment MN in the ratio 2 : 3.
Find:
(i) the coordinates of M and N
(ii) slope of the line MN.
Point P(– 4, 6) divides point A(– 6, 10) and B(m, n) in the ratio 2:1, then find the coordinates of point B
A line intersects the y-axis and x-axis at the points P and Q, respectively. If (2, –5) is the mid-point of PQ, then the coordinates of P and Q are, respectively ______.
Find the coordinates of the point R on the line segment joining the points P(–1, 3) and Q(2, 5) such that PR = `3/5` PQ.
If A and B are (– 2, – 2) and (2, – 4) respectively; then find the co-ordinates of the point P such that `(AB)/(AB) = 3/7`.
