If $P = (1,0), Q = (-1,0)$ and $R= (2,0)$ are three given points, then locus of the points satisfying the relation $SQ^2 + SR^2 = 2SP^2,$ is
- a straight line parallel to X-axis
- a circle passing through the origin
- a circle with the centre at the origin
- a straight line parallel to 7-axis
The Correct Option is D
Solution and Explanation
The correct option is(D): a straight line parallel to 7-axis.
Let the coordinate of S be (x, y)
\(\because \hspace15mm SQ^2 + SR^2 = 2SP^2\)
\(\Rightarrow (x + 1)^2 + y^2 + (x - 2)^2 + y^2 = 2 [(x - 1)^2 + y^2]\)
\(\Rightarrow x^2+ 2x+ 1 + y^2+ x^2-4x+ 4 + y^2=2(x^2- 2 x + 1 + y^2)\)
\(\Rightarrow 2 x + 3 = 0 \Rightarrow x=- \frac{3}{2}\)
Hence, it is a straight line parallel to 7-axis.
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Concepts Used:
Straight lines
A straight line is a line having the shortest distance between two points.
A straight line can be represented as an equation in various forms, as show in the image below:
The following are the many forms of the equation of the line that are presented in straight line-
1. Slope – Point Form
Assume P0(x0, y0) is a fixed point on a non-vertical line L with m as its slope. If P (x, y) is an arbitrary point on L, then the point (x, y) lies on the line with slope m through the fixed point (x0, y0) if and only if its coordinates fulfil the equation below.
y – y0 = m (x – x0)
2. Two – Point Form
Let's look at the line. L crosses between two places. P1(x1, y1) and P2(x2, y2) are general points on L, while P (x, y) is a general point on L. As a result, the three points P1, P2, and P are collinear, and it becomes
The slope of P2P = The slope of P1P2 , i.e.
\(\frac{y-y_1}{x-x_1} = \frac{y_2-y_1}{x_2-x_1}\)
Hence, the equation becomes:
y - y1 =\( \frac{y_2-y_1}{x_2-x_1} (x-x1)\)
3. Slope-Intercept Form
Assume that a line L with slope m intersects the y-axis at a distance c from the origin, and that the distance c is referred to as the line L's y-intercept. As a result, the coordinates of the spot on the y-axis where the line intersects are (0, c). As a result, the slope of the line L is m, and it passes through a fixed point (0, c). The equation of the line L thus obtained from the slope – point form is given by
y – c =m( x - 0 )
As a result, the point (x, y) on the line with slope m and y-intercept c lies on the line, if and only if
y = m x +c