Suppose the pointegral (P(1,1)) is translated to (Q) in the... | Mathematics | SolveeIt

Suppose the point $P(1,1)$ is translated to $Q$ in the direction of y = 2x. If $PQ = 1$ ,then Qis

$(2, 0)$

$(0,2)$

$(\dfrac{√2+1}{√2},\dfrac{√2+1}{√2})$

$(\dfrac{√5+1}{√5},\dfrac{√5+2}{√5})$

$(\dfrac{2+√3}{2},\dfrac{3}{2})$

Answer

$(\dfrac{√5+1}{√5},\dfrac{√5+2}{√5})$

Explanation

Given that:

$y=2x$ , is the equation of the line passing through the points P(1,1)

So , $y−1=2(x−1)$

$⇒y=2x−1$

Let the position of $Q$ be $(x1,y1)$

Since, $Q(x1,y1)$ lies on this line

⇒ $y_1=2x_1−1$ ----------(1)

Also, given P is translated to Q by a unit distance (As PQ=1)

$PQ=1$

$⇒(x_1−1)^2+(y_1−1)^2=1$

$⇒(x_1−1)^2+(2x_1−2)^2=1$ (from equation (1))

$⇒5x_1^2−10x_1+4=0$

Now to get values of $x_1$ from the quadratic equation we can write:

$⇒x_1=\dfrac{10±√20}{10}$

$⇒x_1=\dfrac{5±√5}{5}$
$⇒x_1=1±\dfrac{1}{√5}$

Substitute the above value in equation(1), we get

$⇒y_1=1±\dfrac{2}{√5}$
Hence, the new position of $P$ is $(1±\dfrac{1}{√5},1±\dfrac{2}{√5}) (\text{Ans})$

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