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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

Let \(\mathrm{A}(2,-3)\) and \(\mathrm{B}(-2,1)\) be vertices of a triangle \(\mathrm{ABC}\). If the centroid of this triangle moves on the line \(2 x+3 y=1\), then locus of the vertex \(\mathrm{C}\) is the line (a) \(2 \mathrm{x}+3 \mathrm{y}=9\) (b) \(2 x-3 y=7\) (c) \(3 \mathrm{x}+2 \mathrm{y}=5\) (d) \(3 x-2 y=3\)

For a \(+b+c=0\), the line \(3 \mathrm{ax}+4 \mathrm{by}+\mathrm{c}=0\) passes through the fixed point \(\ldots \ldots .\) (a) \([(1 / 3),-(1 / 4)]\) (b) \([-(1 / 3),(1 / 4)]\) (c) \([(1 / 3),(1 / 4)]\) (d) \([-(1 / 3),-(1 / 4)]\)

If the lengths of perpendicular drawn from the origin to the lines \(x \cos \alpha-y \sin \alpha=\sin 2 a \alpha\) and \(x \sin \alpha+y \cos \alpha=\cos 2 \alpha\) are \(p\) and \(q\) respectively, then \(p^{2}+q^{2}=\ldots \ldots\) (a) 4 (b) 3 (c) 2 (d) 1

The y-intercept of the line \(\mathrm{y}+\mathrm{y}_{1}=\mathrm{m}\left(\mathrm{x}-\mathrm{x}_{1}\right)\) is \(\ldots \ldots\) (a) \(-\left(\mathrm{y}_{1}+\mathrm{mx}_{1}\right)\) (b) \(\mathrm{y}_{1}-\mathrm{mx}_{1}\) (c) \(\left[\left(\mathrm{y}_{1}+\mathrm{mx}_{1}\right) / \mathrm{m}\right]\) (d) None of these

If \(\mathrm{a}_{1}, \mathrm{a}_{2}, \mathrm{a}_{3}\) and \(\mathrm{b}_{1}, \mathrm{~b}_{2}, \mathrm{~b}_{3}\) are in geometric progression and their common ratios are equal, then the points \(\mathrm{A}\left(\mathrm{a}_{1}, \mathrm{~b}_{1}\right)\) \(\mathrm{B}\left(\mathrm{a}_{2}, \mathrm{~b}_{2}\right)\) and \(\mathrm{C}\left(\mathrm{a}_{3}, \mathrm{~b}_{3}\right) \ldots \ldots\) (a) lie on the same line (b) lie on a circle (c) lie on an ellipse (d) None of these
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