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The ratio in which the point P(-3,y) divides the line segment joining the points A(-5,-4) and B(-2,3) is
- a)2: 1
- b)1: 2
- c)-2: 1
- d)2: -1
Correct answer is option 'A'. Can you explain this answer?
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To find the ratio in which the point P(-3,y) divides the line segment AB joining the points A(-5,-4) and B(-2,3), we can use the section formula.
The section formula states that if a point P(x,y) divides a line segment joining two points A(x₁,y₁) and B(x₂,y₂) in the ratio m:n, then the coordinates of P are given by:
x = (mx₂ + nx₁) / (m + n)
y = (my₂ + ny₁) / (m + n)
In this case, we are given that P(-3,y) divides AB. We need to find the value of y and the ratio in which P divides AB.
Let's substitute the given values into the section formula:
x = (-3) = (m(-2) + n(-5)) / (m + n)
y = y = (m(3) + n(-4)) / (m + n)
Simplifying the equations:
-3(m + n) = -2m - 5n
y(m + n) = 3m - 4n
Expanding the equations:
-3m - 3n = -2m - 5n
ym + yn = 3m - 4n
Combining like terms:
-m - 2n = 0
ym + yn = 3m - 4n
Simplifying further:
m + 2n = 0
y = 3m/n - 4
To find the ratio, we can assume a value for m and solve for n. Let's assume m = 2:
2 + 2n = 0
n = -1
So, the ratio m:n is 2:1.
This means that the point P(-3,y) divides the line segment AB in the ratio 2:1.
The section formula states that if a point P(x,y) divides a line segment joining two points A(x₁,y₁) and B(x₂,y₂) in the ratio m:n, then the coordinates of P are given by:
x = (mx₂ + nx₁) / (m + n)
y = (my₂ + ny₁) / (m + n)
In this case, we are given that P(-3,y) divides AB. We need to find the value of y and the ratio in which P divides AB.
Let's substitute the given values into the section formula:
x = (-3) = (m(-2) + n(-5)) / (m + n)
y = y = (m(3) + n(-4)) / (m + n)
Simplifying the equations:
-3(m + n) = -2m - 5n
y(m + n) = 3m - 4n
Expanding the equations:
-3m - 3n = -2m - 5n
ym + yn = 3m - 4n
Combining like terms:
-m - 2n = 0
ym + yn = 3m - 4n
Simplifying further:
m + 2n = 0
y = 3m/n - 4
To find the ratio, we can assume a value for m and solve for n. Let's assume m = 2:
2 + 2n = 0
n = -1
So, the ratio m:n is 2:1.
This means that the point P(-3,y) divides the line segment AB in the ratio 2:1.
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The ratio in which the point P(-3,y) divides the line segment joining the points A(-5,-4) and B(-2,3) isa)2: 1b)1: 2c)-2: 1d)2: -1Correct answer is option 'A'. Can you explain this answer?
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