Question

If the lines $$y = \left( {2 + \sqrt 3 } \right)x + 4$$    and $$y = kx + 6$$   are inclined at an angle $${60^ \circ }$$  to each other, then the value of $$k$$ will be :

A. $$1$$
B. $$2$$
C. $$ - 1$$  
D. $$ - 2$$
Answer :   $$ - 1$$
Solution :
$$\eqalign{ & y = \left( {2 + \sqrt 3 } \right)x + 4......\left( 1 \right) \cr & y = k\alpha + 6......\left( 2 \right) \cr & {\text{By equation }}\left( 1 \right) \Rightarrow {m_1} = \left( {2 + \sqrt 3 } \right)\,; \cr & {\text{By equation }}\left( 2 \right) \Rightarrow {m_2} = k \cr & \therefore \,\tan \,{60^ \circ } = \frac{{{m_1} - {m_2}}}{{1 + {m_1}{m_2}}} \Rightarrow k = - 1 \cr} $$

Releted MCQ Question on
Geometry >> Straight Lines

Releted Question 1

The points $$\left( { - a, - b} \right),\left( {0,\,0} \right),\left( {a,\,b} \right)$$     and $$\left( {{a^2},\,ab} \right)$$  are :

A. Collinear
B. Vertices of a parallelogram
C. Vertices of a rectangle
D. None of these
Releted Question 2

The point (4, 1) undergoes the following three transformations successively.
(i) Reflection about the line $$y =x.$$
(ii) Translation through a distance 2 units along the positive direction of $$x$$-axis.
(iii) Rotation through an angle $$\frac{p}{4}$$ about the origin in the counter clockwise direction.
Then the final position of the point is given by the coordinates.

A. $$\left( {\frac{1}{{\sqrt 2 }},\,\frac{7}{{\sqrt 2 }}} \right)$$
B. $$\left( { - \sqrt 2 ,\,7\sqrt 2 } \right)$$
C. $$\left( { - \frac{1}{{\sqrt 2 }},\,\frac{7}{{\sqrt 2 }}} \right)$$
D. $$\left( {\sqrt 2 ,\,7\sqrt 2 } \right)$$
Releted Question 3

The straight lines $$x + y= 0, \,3x + y-4=0,\,x+ 3y-4=0$$         form a triangle which is-

A. isosceles
B. equilateral
C. right angled
D. none of these
Releted Question 4

If $$P = \left( {1,\,0} \right),\,Q = \left( { - 1,\,0} \right)$$     and $$R = \left( {2,\,0} \right)$$  are three given points, then locus of the point $$S$$ satisfying the relation $$S{Q^2} + S{R^2} = 2S{P^2},$$    is-

A. a straight line parallel to $$x$$-axis
B. a circle passing through the origin
C. a circle with the centre at the origin
D. a straight line parallel to $$y$$-axis

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Straight Lines


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