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Chapter 14: Problem 1233

The angle between the lines \(x \cos 85^{\circ}+y \sin 85^{\circ}=1\) and \(x \cos 40^{\circ}+y \sin 40^{\circ}=2\) is : (a) \(90^{\circ}\) (b) \(80^{\circ}\) (c) \(125^{\circ}\) (d) \(45^{\circ}\)

Short Answer

Expert verified
The angle between the lines is (b) \(80^{\circ}\).

Step by step solution

01

Find Direction Ratios of Line 1

Let \(A_1(x_1, y_1)\) be a point on Line 1 such that \(x_1 \cos 85^{\circ} + y_1 \sin 85^{\circ} = 1\). We can find the direction ratios of Line 1 by considering the vector \(\vec{A_1}\).
02

Find Direction Ratios of Line 2

Let \(A_2(x_2, y_2)\) be a point on Line 2 such that \(x_2 \cos 40^{\circ} + y_2 \sin 40^{\circ} = 2\). Similarly, we can find the direction ratios of Line 2 by considering the vector \(\vec{A_2}\).
03

Calculate the Dot Product and Angle between Direction Ratios

Now that we have the direction of vectors for both lines, we can find the angle between them. Let \(\vec{v_1}\) and \(\vec{v_2}\) be the direction vectors for Line 1 and Line 2 respectively. The angle between these two lines, \(\theta\), can be found using the dot product formula: \[\cos \theta = \frac{\vec{v_1} \cdot \vec{v_2}}{||\vec{v_1}|| \; ||\vec{v_2}||}\] Where \(\vec{v_1} \cdot \vec{v_2}\) is the dot product of the direction vectors and ||\(\vec{v_1}\)|| and ||\(\vec{v_2}\)|| are the magnitudes of the direction vectors.
04

Calculate the Values and Choose the Correct Answer

After finding the value of the angle \(\theta\), we can compare it to the given options and choose the one that matches. (a) \(90^{\circ}\) (b) \(80^{\circ}\) (c) \(125^{\circ}\) (d) \(45^{\circ}\)

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Most popular questions from this chapter

If the line \((a+1) x+\left(a^{2}-a-2\right) y+a=0\) is parallel to \(Y-\) axis, then \(\mathrm{a}=\ldots \ldots\) (a) - 1 (b) 2 (c) 3 (d) 1

For any values of \(\mathrm{p}\) and \(\mathrm{q}\), the line \((p+2 q) x+(p-3 q) y-p+q\) passes through which fixed point? (a) \([(3 / 2),(5 / 2)]\) (b) \([(2 / 5),(2 / 5)]\) (c) \([(3 / 5) /(3 / 5)]\) (d) \([(2 / 5),(3 / 5)]\)

If the \(\mathrm{y}\) -intercept of the perpendicular bisector of the segment obtained by joining \(\mathrm{P}(1,4)\) and \(\mathrm{Q}(\mathrm{k}, 3)\) is \(-4\) then \(\mathrm{k}=\ldots \ldots\) (a) 1 (b) 2 (c) \(-2\) (d) \(-4\)

The foot of perpendicular drawn from \((2,3)\) to the line \(4 x-5 y-34=0\) is ..... (a) \((6,-2)\) (b) \([(246 / 41),(82 / 41)]\) (c) \((-6,2)\) (d) None of these

If the equation of base of an equilateral triangle is \(2 \mathrm{x}-\mathrm{y}=1\) and the vertex is \((-1,2)\), then the length of the side of the triangle is (a) \(\sqrt{(20 / 3)}\) (b) \((2 / \sqrt{1} 5)\) (c) \(\sqrt{(8 / 15)}\) (d) \(\sqrt{(15 / 2)}\)
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