Question

12. A line forms a triangle with co-ordinate axes. If the area of this triangle is 54√3 square units and the perpendicular drawn from the origin to the line makes an angle of 60° with the x-axis, find the equation of the line. solve this sum
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Answer 1

Answer:

forms a triangle with coordinate axes which clearly means that the triangle will be a right-angled triangle. The Area of the triangle is also given so we need to calculate the height and base to form the required equation.

Complete step-by-step answer:

In the question, it is given that a line forms a triangle of area 543–√ sq. with coordinate axes,

So, let us take a line AB

as shown in figure which forms a triangle of area 543–√ sq. with coordinate axes.

Since, we can see that point A

lies on x-axis, therefore the y-coordinate of A

will be 0

So, A=(a,0)

Also, we can see that point B

lies on y-axis, therefore the x-coordinate of B

will be 0

So, B=(0,b)

As we know, area of a triangle =12×height×base

Now, in △AOB

, height =b

and base =a

Area of △AOB=12×b×a

And Area of △AOB

= 543–√

Therefore, 543–√=12ab

Or, ab=1083–√

--- (1)

Now, the perpendicular drawn from the origin to the line makes an angle of 60∘

with x-axis as shown in figure.

Let the length of the perpendicular drawn from origin to line be P

.

Also, we know that in a right-angled triangle, cosθ=BaseHypotenuse

cos60∘=Pa

and cos60∘=12

Or, a=Pcos60∘=2P

And, cos30∘=Pb

and cos30∘=3–√2

Or, b=Pcos30∘=2P3–√

Now, put values of a

and b

in equation (1)

,

2P×2P3–√=1083–√

Or, 4P23–√=1083–√

Or, P2=108×3

P=±18

But we can only take P=18

because the triangle is in the1st

quadrant.

So, P=18

And, a=2P=36

, b=2P3–√=363–√=123–√

The intercept form of the equation of the straight line is xa+yb=1

So, the equation of the line becomes x36+y123–√=1

Hence, the equation of line comes out to be x+3–√y−36=0

Step-by-step explanation:

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Answer 2

Answer:

0 is the correct Answer

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