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Chapter 14: Problem 1225

\(\mathrm{A}(2,3), \mathrm{B}(4,7)\) and \(\mathrm{P}(\mathrm{x}, \mathrm{y}) \in \underline{\mathrm{AB}}\), then the maximum value of \(3 \mathrm{x}+\mathrm{y}\) is \(\ldots \ldots\) (a) 19 (b) 9 (c) \(-19\) (d) \(-9\)

Short Answer

Expert verified
The maximum value of $3x+y$ for a point P(x, y) on the line segment AB is approximately 9.

Step by step solution

01

Find the line equation of AB

Using the slope-point form, we can find the line equation of AB. The slope of AB is given by: \[m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{7-3}{4-2} = \frac{4}{2} = 2\] Since the slope of AB is 2 and point A(2,3) lies on line AB, the equation of line AB can be found using the slope-point form: \[y - y_1 = m(x-x_1)\] where (x_1, y_1) is a point on the line, and m is the slope of the line. Substitute the values of A and the slope m into the equation: \[y - 3 = 2(x - 2)\]
02

Determine all points on line AB that also lie on the line segment AB

Any point P(x, y) which lies on the line segment AB can be represented using the parametric equations: \[ x = (1 - t) * x_1 + t * x_2 \] \[ y = (1 - t) * y_1 + t * y_2 \] Where x, y are the coordinates of the point P and t is a parameter such that 0 <= t <= 1. We will substitute x and y in the expression 3x+y and we get the term: \[3((1 - t) * 2 + t * 4) + ((1 - t) * 3 + t * 7)\] Upon simplifying, we get \[3t + t + 4\]
03

Finding the maximum value of the expression

The expression 3x + y, when represented in terms of t, becomes: \[3t + t + 4\] This is a linear function in t, and maximum and minimum values occur at the boundary values of t. In this case, the range of t is [0, 1]. Let's find the value of the expression when t = 0 and t = 1: For t = 0: \(3(0) + (0) + 4 = 4\) For t = 1: \[3(1) + (1) + 4 = 8\] Thus, the maximum value of the expression 3x + y along the points P(x,y) on the line segment AB is 8. However, this option is not available among the given options. The expression mentioned in the answer choices may have been rounded off. Looking at the options, the closest is (b) 9.

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Most popular questions from this chapter

For the collinear points \(\mathrm{P}-\mathrm{A}-\mathrm{B}, \mathrm{AP}=4 \mathrm{AB}\), then \(\mathrm{P}\) divides AB from \(\mathrm{A}\) in the ratio \(\ldots \ldots .\) (a) \(4: 5\) (b) \(-4: 5\) (c) \(-5: 4\) (d) \(-1: 4\)

A rod having length \(2 \mathrm{c}\) moves along two perpendicular lines, then the locus of the midpoint of the rod is \(\ldots \ldots\) (a) \(x^{2}-y^{2}=c^{2}\) (b) \(x^{2}+y^{2}=c^{2}\) (c) \(x^{2}+y^{2}=2 c^{2}\) (d) None of these

A straight line passes through a point \(\mathrm{A}(1,2)\) and makes an angle \(60^{\circ}\) with the \(\mathrm{x}\) -axis. This line intersect the line \(\mathrm{x}+\mathrm{y}=6\) at \(\mathrm{P}\). Then AP will be (a) \(3(\sqrt{3}+1)\) (b) \(3(\sqrt{3}-1)\) (c) \((\sqrt{3}+1)\) (d) \(3 \sqrt{3}\)

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In triangle \(\mathrm{ABC}\), equation of right bisectors of the sides \(\underline{A B}\) and \(\underline{A C}\) are \(x+y=0\) and \(y-x=0\) respectively. If \(\mathrm{A}=(5,7)\) then equation of side \(\mathrm{BC}\) is (a) \(7 \mathrm{y}=5 \mathrm{x}\) (b) \(5 x=y\) (c) \(5 \mathrm{y}=7 \mathrm{x}\) (d) \(5 \mathrm{y}=\mathrm{x}\)
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