Download App

The Straight Lines — MATHS STD 11 Science — Question

Rajasthan BoardEnglish MediumSTD 11 ScienceMATHSThe Straight Lines5 Marks
Question
Prove that the lines $\text{y}=\sqrt{3}\text{x}+1,\text{y}=4$ and $\text{y}=-\sqrt{3}\text{x}+2$ form an equilateral triangle.
✓

Answer

The given equations are as follows:
$\text{y}=\sqrt3\text{x}+1\ ...(1)$
$\text{y}=4\ ...(2)$
$\text{y} = -\sqrt{3}\text{x}+2\ ...(3)$
In triangle ABC, let equation (1), (2) and (3) represent the sides AB, BC and CA, respectively.
Solving (1) and (2) 
$\text{x}= \sqrt3,\text{y}=4$
Thus, AB and BC intersect at $\text{B}\big(\sqrt3,4\big)$
Solving (1) and (3)
$\text{x} = \frac{1}{2\sqrt3}, \text{y} = \frac{3}{2}$
Thus, AB and CA intersect at $\text{A}\big(\frac{1}{2\sqrt3},\frac{3}{2}\big).$
Similarly, solving (2) and (3)
$\text{x} = -\frac{2}{\sqrt3}, \text{y}= 4$
Thus, BC and AC intersect at $\text{C}\big(-\frac{2}{\sqrt{3}},4\big).$
Now, we have:
$\text{AB}=\sqrt{\Big(\frac{1}{2\sqrt{3}}-\sqrt{3}\Big)^2+\Big(\frac{3}{2}-4\Big)^2}=\frac{5}{\sqrt3}$
$\text{BC}=\sqrt{\Big(\sqrt{3}+\frac{2}{\sqrt3}\Big)^2+\Big(4-4\Big)^2}=\frac{5}{\sqrt3}$
$\text{AC}=\sqrt{\Big(\frac{1}{2\sqrt3}+\frac{2}{\sqrt3}\Big)^2+\Big(\frac{3}{2}-4\Big)^2}=\frac{5}{\sqrt3}$
Hence, the given lines form an equilateral triangle.

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "5 Marks Questions" from The Straight Lines→The Straight Lines — all question types→MATHS STD 11 Science question papers→Rajasthan Board STD 11 Science question papers→Rajasthan Board question paper generator→

Similar questions

Evaluate the following limit:
$\lim\limits_{\text{x}\rightarrow0}\frac{\sin2\text{x}(\cos3\text{x}-\cos\text{x})}{\text{x}^3}$
→
Prove that the points (2, -1), (0, 2), (2, 3) and (4, 0) are the coordinates of the vertices of a parallelogram and find the angle between its diagonals.
→
In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
→
 Prove the following:
$\sin2\text{x}+2\sin4\text{x}+\sin6\text{x}=4\cos^2\text{x}\sin4\text{x}$
→
In a potato race 20 potatoes are placed in a line at intervals of 4 meters with the first potato 24 metres from the starting point. A contestant is required to bring the potatoes back to the starting place one at a time. How far would he run in bringing back all the potatoes?
→
$\frac{\sqrt{\sin\text{A}}-\sqrt{\sin\text{B}}}{\sqrt{\sin\text{A}}+\sqrt{\sin\text{B}}}=\frac{\text{a + b}-2\sqrt{\text{ab}}}{\text{a}-\text{b}}$
→
Find the derivative of x sinx from first principle.
→
Prove that:
$\sin\frac{\text{x}}{2}\sin\frac{7\text{x}}{2}+\sin\frac{3\text{x}}{2}\sin\frac{11\text{x}}{2}=\sin2\text{x}\sin5\text{x}.$
→
Prove the following by the principle of mathematical induction:
11n+2 + 122n+1 is divisible of 133 for all $\text{n}\in\text{N}$
→
Differentiate the following from first principle

$\cos\sqrt{\text{x}}$

→