The angle made by the line root{3}x+y+6=0, with y-axis is

Question

The angle made by the line  \sqrt{3}x+y+6=0 , with y-axis is

A. 60 degrees
B. 30 degrees
C. 45 degrees
D. 120 degrees
E. None of the above

Bunuel · Accepted Answer

We have the line y=-\sqrt{3}*x-6 . untitled.PNG Find the x-intercept of the line (x-intercept of a line the value of x for y=0, the point where the line crosses X-axis): y=0 --> x=-2\sqrt{3} ; Find the y-intercept of the line (y-intercept of a line the value of y for x=0, the point where the line crosses Y-axis): x=0 --> y=-6 ; In the right triangle created by the given line and the axis the ratio of lengths of the legs is 2\sqrt{3}:6=1:\sqrt{3} so we have 30°, 60°, and 90° right triangle (in a right triangle where the angles are 30°, 60°, and 90° the sides are always in the ratio 1 : \sqrt{3}: 2 . Notice that the smallest side (1) is opposite the smallest angle (30°), and the longest side (2) is opposite the largest angle (90°)). Now, the angle at y-axis is opposite the smallest side 2\sqrt{3} so it equals to 30°. Answer: B. Or you can notice that as the slope, which is th ratio of the rise over run is equal to -\sqrt{3} then the ratio of the length of the leg at y-axis over the length of the leg at x-axis will be \sqrt{3} , which again tells us that we have 30°, 60°, and 90° right triangle. For more on this check Triangles and Coordinate Geometry chapters of Math Book: math-triangles-87197.html and math-coordinate-geometry-87652.html Hope it's clear.

mailnavin1 · Answer

that was smart. I tried to use m = tan(degree), Is this wrong. If no can you please explain how to compute using this.

Bunuel · Answer

Every GMAT geometry question can be solved without trigonometry, so you don't need trigonometry for GMAT.

But yes,  slope=m=tan(\alpha) , where  \alpha  is the angle line makes with positive x-axis, so as  m=-\sqrt{3}=tan(\alpha)  -->  \alpha=120^{\circ}  --> angle at y-axis will be 120-90=30.

tfincham86 · Answer

Good explanation.

Thanks Bunuel

hermit84 · Answer

This question is incomplete, for answer to be 30 degrees, question should ask the smaller angle between line and y-axis. Otherwise answer can be either 150 degrees or 30 degrees.

Bunuel · Answer

Though I see your point, still only one answer from 150 and 30 is listed among answer choices, so we can safely choose 30 as correct one.

jullysabat · Answer

Hello Bunnel,

Can you pls explain how did draw the graph...
How did you take the points of x and y..

Bunuel · Answer

y=mx+b is called point-intercept form of equation of a line (in our case it's y=-\sqrt{3}*x-6 ), where m is the slope of the line; b is the y-intercept of the line; x is the independent variable of the function y . To draw a graph of a line you need two points (any two distinct points (x,y) which satisfy the equation of a line) and then you just draw a line passing these points. Usually X and Y intercepts are best choices for the points. X-intercept is the point where a line (a graph) crosses the x-axis. So it's the point on x-axis, any point on x-axis has y-coordinate equal to zero, which means that X-intercept is the point (x,0) - the value of x when y=0 : y=mx+b --> 0=mx+b --> x=-\frac{b}{m} . So X-intercept of a line y=mx+b is x=-\frac{b}{m} ; Y-intercept is the point where a line (a graph) crosses the y-axis. So it's the point on y-axis, any point on y-axis has x-coordinate equal to zero, which means that Y-intercept is the point (0,y) - the value of y when x=0 : y=mx+b --> y=m*0+b --> y=b . So Y-intercept of a line y=mx+b is y=b . In our case for the line given by y=-\sqrt{3}*x-6 : X-intercept will be (-2\sqrt{3}, \ 0) , as x-intercept of a line is the value of x for y=0, the point where the line crosses X-axis: y=0 --> x=-2\sqrt{3} ; Y-intercept will be (0, \ -6) , as y-intercept of a line is the value of y for x=0, the point where the line crosses Y-axis: x=0 --> y=-6 . Check Coordinate Geometry chapter for more: math-coordinate-geometry-87652.html Hope it's clear.

jullysabat · Answer

Yeah its clear now...
Thanks..

144144 · Answer

very nice work Bunuel. thanks for putting so much effort.

vjsharma25 · Answer

But why the tan(@) formula works on this. As Y axis has a slope = infinity. So this formula doesn't work and i had chosen last option which says "None of the above". I know this option may not be there in GMAT but why can't we apply a particular formula to reach the same result.
 
 Tan(@) = |(m1 - m2)/(1+m1*m2)|

Where m1 and m2 are the slopes of two lines and "@" is angle between these line.

guerrero25 · Answer

Hi Bunuel , Sorry to dig in an old post . I am unable to follow your second approach properly (i.e how did you find the ratio length of y to length of x as sqrt3) . Could you please elaborate it more .

thanks !

vinnik · Answer

The explanation of this question is fantastic and one can quickly solve these kind of questions after understanding it.
The only concept that i am not able to understand is how  2\sqrt{3}:6  turn into  1:\sqrt{3}

I will really appreciate if anyone can help me in understanding this concept.

Thanks & Regards
Vinni

ygdrasil24 · Answer

Vini,

A line of the form y=mx+b, has a slope 'm'

Convert the given equation in this form to get y=-rt3x -6.

Since quesn deals in slope you need not calculate intercepts, clearly slope is negative and is tan(rt30 which is 60 degree. It lies in 3rd quadrant
Since slope is defined as what angle a line makes with x -axis, the angle with y axis is 90-60 = 30.

If you want to know about intercepts, the above eqn cut x axis when y=0 rt?
So when y=0, x = -6/rt3 = 3* 2 / rt3 = -2rt3.
Also when x=0, y=-6.

Hope it helped.

madn800 · Answer

Also for a 45-45-90 angled right angled triangle the corresponding sides are in the ratio 1: 1:  \sqrt{2}

madn800 · Answer

"The explanation of this question is fantastic and one can quickly solve these kind of questions after understanding it.
The only concept that i am not able to understand is how 2 \sqrt{3} :6 turn into 1: \sqrt{3} "

here goes
first cancel with 2 on either side of the ratio
 \sqrt{3}  : 3
on the right 3 can be written as  \sqrt{3} * \sqrt{3} 
  \sqrt{3}  :  \sqrt{3} *  \sqrt{3} 
cancel  \sqrt{3}  
you get 1:  \sqrt{3}

madn800 · Answer

Another idea: Observe buneul's diagram. The coordinates on the -ve x-axis and -ve y-axis are (-2 \sqrt{3} ,0) and (0,-6). Now find the distance between them using the formula  \sqrt{x^2+y^2} . We get 4 \sqrt{3} =2*2 \sqrt{3} . Now we know that the ratio of sides for a 30-60-90 triangle is x: \sqrt{3} x:2x. From distance calculation above we can see that the following ratio holds perfectly good 30:60:90=2 \sqrt{3} :  \sqrt{3} *(2 \sqrt{3} ): 2*(2 \sqrt{3} ). Comparing we can say that side opposite to 2 \sqrt{3}  measures 30 degrees.

mvictor · Answer

this one can be solved easily just by picturing the graph...

rewrite equation: y= - sqrt(3)*x -6
when x=0, y=-6. 
when x=sqrt(3), y=-9
so 3 down the y axis, and 1 to the right of x axis.
we can draw a right triangle, in which 2 legs are known. sqrt(3) & 3.
now..hypotenuse^2 is 3^2 + sqrt(3)^2 = 12. so hypotenuse is 2*sqrt(3)

clearly not 45 and 120
we know that sides of a triangle with angles 30-60-90 must be in this proportion: x:x*sqrt(3):2x.
x=sqrt(3); 2x=2*sqrt(3), and x*sqrt(3) = 3. we have a 30-60-90 triangle, and the angle formed at the intersection with y axis is 30 degrees.

bumpbot · Answer

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The angle made by the line root{3}x+y+6=0, with y-axis is

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