Suppose the point \(P(1,1)\) is translated to \(Q\) in the direction of y = 2x. If \(PQ = 1\),then Qis
\((2, 0)\)
\((0,2)\)
\((\dfrac{√2+1}{√2},\dfrac{√2+1}{√2})\)
\((\dfrac{√5+1}{√5},\dfrac{√5+2}{√5})\)
\((\dfrac{2+√3}{2},\dfrac{3}{2})\)
The Correct Option is D
Solution and 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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Concepts Used:
General Equation of a Line
Equation of Straight Line Formula:
A straight line is a figure created when two points A (x1, y1) and B (x2, y2) are connected with a minimum distance between them, and both the ends are extended to immensity (infinity). With variables x and y, the standard form of a linear equation is: ax + by = c, where a, b, and c are constants and x, and y are variables.
Point Slope Form:
The equation of a straight line whose slope is m and passes through a point (x1, y1) is formed or created using the point-slope form. The equation of the point-slope form is:
y - y1 = m (x - x1),
where (x, y) = an arbitrary point on the line.
The Slope of a Line
A slope of a line is the conversion in y coordinate w.r.t. the conversion in x coordinate.
The net change in the y-coordinate is demonstrated by Δy and the net change in the x-coordinate is demonstrated by Δx.
Hence, the change in y-coordinate w.r.t. the change in x-coordinate is given by,
\(m = \frac{\text{change in y}}{\text{change in x}} = \frac{Δy}{Δx}\)
Where, “m” is the slope of a line.
The slope of the line can also be shown by
\(tan θ = \frac{Δy}{Δx}\)
Read More: Slope Formula
The slope of a Line Equation:
The equation for the slope of a line and the points are known to be a point-slope form of the equation of a straight line is given by:
\(y-y_1=m(x-x_1)\)
As long as the slope-intercept form the equation of the line is given by:
\(y = mx + b\)
Where, b is the y-intercept.