Mathematics, 19.12.2020 08:30 s9066115

$\underline{ \underline{ \text{question}}} :$ Find the equation of straight line which cuts off an intercept 2 from the Y - axis and whose perpendicular distance from the origin is 1.

Find the equation of straight line which cuts off an intercept 2 from the Y - axis and whose perpe

Answers: 2

Show answers

Answers

Answer from: vivian2020

$y=-\sqrt{3}x+2$

Step-by-step explanation:

We want to find the equation of a straight line that cuts off an intercept of two from the y-axis and whose perpendicular distance from the origin is one.

Let Point M be (x, y). As we know, Point R will be (0, 2) and Point O (the origin) will be (0, 0).

First, we can use the distance formula to determine values for M. The distance formula is given by:

$\displaystyle d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Since we know that the distance between O and M is one, d = 1:

And we will let M(x, y) be (x₂, y₂) and O(0, 0) be (x₁, y₁). So:

$\displaystyle 1=\sqrt{(x-0)^2+(y-0)^2}$

Simplify:

$1=\sqrt{x^2+y^2}$

We can solve for y. Square both sides:

1=x^2+y^2

Rearranging gives:

y^2=1-x^2

Take the square root of both sides. Since M is in the first quadrant, we only need to worry about the positive case. Therefore:

$y=\sqrt{1-x^2}$

So, Point M is now given by (we substitute the above equation for y):

$M(x,\sqrt{1-x^2})$

We know that Segment OM is perpendicular to Line RM.

Therefore, their slopes will be negative reciprocals of each other.

So, let’s find the slope of each segment/line. We will use the slope formula given by:

$\displaystyle m=\frac{y_2-y_1}{x_2-x_1}$

Segment OM:

For OM, we have two points: O(0, 0) and M(x, √(1-x²)). So, the slope will be:

$\displaystyle m_{OM}=\frac{\sqrt{1-x^2}-0}{x-0}=\frac{\sqrt{1-x^2}}{x}$

Line RM:

For RM, we have the two points R(0, 2) and M(x, √(1-x²)). So, the slope will be:

$\displaystyle m_{RM}=\frac{\sqrt{1-x^2}-2}{x-0}=\frac{\sqrt{1-x^2}-2}{x}$

Since their slopes are negative reciprocals of each other, this means that:

$m_{OM}=-(m_{RM})^{-1}$

Substitute:

$\displaystyle \frac{\sqrt{1-x^2}}{x}=-\Big(\frac{\sqrt{1-x^2}-2}{x}\Big)^{-1}$

Now, we can solve for x. Simplify:

$\displaystyle \frac{\sqrt{1-x^2}}{x}=\frac{x}{2-\sqrt{1-x^2}}$

Cross-multiply:

$x(x)=\sqrt{1-x^2}(2-\sqrt{1-x^2})$

Distribute:

$x^2=2\sqrt{1-x^2}-(\sqrt{1-x^2})^2$

Simplify:

$x^2=2\sqrt{1-x^2}-(1-x^2)$

Distribute:

$x^2=2\sqrt{1-x^2}-1+x^2$

So:

$0=2\sqrt{1-x^2}-1$

Adding 1 and then dividing by 2 yields:

$\displaystyle \frac{1}{2}=\sqrt{1-x^2}$

Then:

$\displaystyle \frac{1}{4}=1-x^2$

Therefore, the value of x is:

$\displaystyle \begin{aligned}\frac{1}{4}-1&=-x^2\\-\frac{3}{4}&=-x^2\\ \frac{3}{4}&=x^2\\ \frac{\sqrt{3}}{2}&=x\end{aligned}$

Then, Point M will be:

$\begin{aligned} \displaystyle M(x,\sqrt{1-x^2})&=M\left(\frac{\sqrt{3}}{2}, \sqrt{1-\left(\frac{\sqrt{3}}{2}\right)^2}\right)\\ \\ M&=\left(\frac{\sqrt3}{2},\frac{1}{2}\right)\end{aligned}$

Therefore, the slope of Line RM will be:

$\displaystyle \begin{aligned}m_{RM}&=\frac{\dfrac{1}{2}-2}{\dfrac{\sqrt{3}}{2}-0} \\ \\ &=\frac{\dfrac{-3}{2}}{\dfrac{\sqrt{3}}{2}}\\ \\ &=-\frac{3}{\sqrt3}\\ \\&=-\sqrt3\end{aligned}$

And since we know that R is (0, 2), R is the y-intercept of RM. Then, using the slope-intercept form:

y=mx+b

We can see that the equation of Line RM is:

$y=-\sqrt{3}x+2$

Answer from: ayoismeisjjjjuan

the answer for this question is the picture hope this helps

Find the equation of straight line which cuts off an intercept 2 from the Y - axis and whose perpen

Answer from: Quest

A. p = 2(l + w)p/2 - l = ww = p/2 - l b. while isolating the variable w in the equation, i first divided both sides by 2 and then i subtracted l from both sides. i hope this .

Answer from: Quest

step-by-step explanation:

de-gf

hj=3x+20

ji=15x-64

Another question on Mathematics

Mathematics, 21.06.2019 20:00

What is the x-intercept and y-intercept of the equation 2x+1. 2o

Answers: 2

Answer

Mathematics, 21.06.2019 20:20

One of every 20 customers reports poor customer service on your company’s customer satisfaction survey. you have just created a new process that should cut the number of poor customer service complaints in half. what percentage of customers would you expect to report poor service after this process is implemented? 1.) 5% 2.) 10% 3.) 2% 4.) 2.5%

Answers: 1

Answer

Mathematics, 21.06.2019 22:40

What rotation was applied to triangle def to create d’e’f’?

Answers: 2

Answer

Mathematics, 21.06.2019 23:00

Rob filled 5 fewer plastic boxes with football cards than basketball cards. he filled 11 boxes with basketball cards.

Answers: 1

Answer

You know the right answer?

Find the equation of straight line which cuts off an intercept 2 from the Y - axis and whose perpen...