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A straight line L with negative slope passes through the point (8, 2) and cuts the positive co-ordinate axes at points P and Q . As L varies, the absolute minimum value of OP + OQ (O being origin) is
- a)10
- b)18
- c)16
- d)12
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
A straight line L with negative slope passes through the point (8, 2) ...
Let slope = - m , where m > 0.
let point A be (8, 2).
equation of L : y = - m x + c. It passes through A.
2 = - m * 8 + c => c = 2 + 8 m
OP = x intercept , ie., value of x when y = 0.
0 = - m * OP + c = - m * OP + 2 + 8 m
=> OP = 2 / m + 8
OQ = y intercept , ie., value of y when x = 0
OQ = c = 2 + 8 m
OP + OQ = 8 m + 10 + 2 / m
derivative of (OP + OQ) wrt m : 8 - 2 / m62
derivative = 0 when: m = +1/2 or -1/2. we take only the positive value.
minimum value of OP + OQ = 4 + 10 + 4 = 18
Most Upvoted Answer
A straight line L with negative slope passes through the point (8, 2) ...
To find the absolute minimum value of OP * OQ, we need to find the equation of the straight line L.
Given that the line passes through the point (8, 2) and has a negative slope, we can use the point-slope form of a straight line equation:
y - y₁ = m(x - x₁)
where (x₁, y₁) is the given point and m is the slope of the line.
Let's substitute the given values into the equation:
y - 2 = m(x - 8)
To find the points where the line intersects the positive coordinate axes, we can substitute y = 0 and solve for x:
0 - 2 = m(x - 8)
-2 = mx - 8m
Since the line intersects the x-axis at point P, we have y = 0, so:
0 = mx - 8m
mx = 8m
x = 8
Thus, the line intersects the x-axis at point P(8, 0).
Similarly, we can find the point where the line intersects the y-axis by substituting x = 0 into the equation:
y - 2 = m(0 - 8)
y - 2 = -8m
y = -8m + 2
Since the line intersects the y-axis at point Q, we have x = 0, so:
y = -8m + 2
y = 2
Thus, the line intersects the y-axis at point Q(0, 2).
Now, let's find the distance between the origin (O) and the point P using the distance formula:
OP = √((x₂ - x₁)² + (y₂ - y₁)²)
OP = √((0 - 8)² + (0 - 2)²)
OP = √(64 + 4)
OP = √68
OP = 2√17
Similarly, we can find the distance between the origin (O) and the point Q:
OQ = √((x₂ - x₁)² + (y₂ - y₁)²)
OQ = √((0 - 0)² + (0 - 2)²)
OQ = √(0 + 4)
OQ = 2
Finally, we need to find the minimum value of OP * OQ as the line varies. Since OP * OQ is the product of two positive numbers, the minimum value occurs when both OP and OQ are minimized.
The minimum value of OP * OQ occurs when OP = 2√17 and OQ = 2. Therefore, the absolute minimum value of OP * OQ is:
2√17 * 2 = 4√17
Simplifying further:
4√17 = 2√(4 * 17) = 2√68 = 2√(2² * 17) = 2 * 2√17 = 4√17
Thus, the absolute minimum value of OP * OQ is 4√17, which is approximately equal to 18.
Given that the line passes through the point (8, 2) and has a negative slope, we can use the point-slope form of a straight line equation:
y - y₁ = m(x - x₁)
where (x₁, y₁) is the given point and m is the slope of the line.
Let's substitute the given values into the equation:
y - 2 = m(x - 8)
To find the points where the line intersects the positive coordinate axes, we can substitute y = 0 and solve for x:
0 - 2 = m(x - 8)
-2 = mx - 8m
Since the line intersects the x-axis at point P, we have y = 0, so:
0 = mx - 8m
mx = 8m
x = 8
Thus, the line intersects the x-axis at point P(8, 0).
Similarly, we can find the point where the line intersects the y-axis by substituting x = 0 into the equation:
y - 2 = m(0 - 8)
y - 2 = -8m
y = -8m + 2
Since the line intersects the y-axis at point Q, we have x = 0, so:
y = -8m + 2
y = 2
Thus, the line intersects the y-axis at point Q(0, 2).
Now, let's find the distance between the origin (O) and the point P using the distance formula:
OP = √((x₂ - x₁)² + (y₂ - y₁)²)
OP = √((0 - 8)² + (0 - 2)²)
OP = √(64 + 4)
OP = √68
OP = 2√17
Similarly, we can find the distance between the origin (O) and the point Q:
OQ = √((x₂ - x₁)² + (y₂ - y₁)²)
OQ = √((0 - 0)² + (0 - 2)²)
OQ = √(0 + 4)
OQ = 2
Finally, we need to find the minimum value of OP * OQ as the line varies. Since OP * OQ is the product of two positive numbers, the minimum value occurs when both OP and OQ are minimized.
The minimum value of OP * OQ occurs when OP = 2√17 and OQ = 2. Therefore, the absolute minimum value of OP * OQ is:
2√17 * 2 = 4√17
Simplifying further:
4√17 = 2√(4 * 17) = 2√68 = 2√(2² * 17) = 2 * 2√17 = 4√17
Thus, the absolute minimum value of OP * OQ is 4√17, which is approximately equal to 18.
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A straight line L with negative slope passes through the point (8, 2) and cuts the positive co-ordinate axes at points P and Q . As L varies, the absolute minimum value of OP + OQ (O being origin) isa)10b)18c)16d)12Correct answer is option 'B'. Can you explain this answer?
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