Advertisements
Advertisements
Question
Write the coordinates of a point on X-axis which is equidistant from the points (−3, 4) and (2, 5).
Advertisements
Solution
The distance d between two points `(x_1 , y_1 ) ` and `(x_2 , y_ 2)` is given by the formula
`d = sqrt( (x_1 - x_2 )^2 + (y_1 - y_2 )^2)`
Here we are to find out a point on the x−axis which is equidistant from both the points
A(−3, 4) and B(2, 5).
Let this point be denoted as C(x, y).
Since the point lies on the x-axis the value of its ordinate will be 0. Or in other words we have y = 0 .
Now let us find out the distances from ‘A’ and ‘B’ to ‘C’
`Ac = sqrt( ( - 3- x)^2 + (4 - y )^2)`
`= sqrt((-3-x)^2 + (4 - 0 )^2))`
`AC = sqrt((-3-x)^2 + ( 4)^2`
`BC= sqrt((2 -x)^2 + (5 -y)^2)`
`= sqrt((2 -x)^2 + ( 5 - 0)^2)`
`BC = sqrt((2-x)^2 + (5)^2)`
We know that both these distances are the same. So equating both these we get,
AC = BC
`sqrt((-3-x)^2 + (4)^2 ) = sqrt((2 - x)^2 + (5)^2)`
Squaring on both sides we have,
`(-3-x)^2 + (4)^2 = (2 -x)^2 + (5)^2`
`9 +x^2 + 6x + 16 = 4 +x^2 - 4x + 25`
`10x = 4`
` x = 2/5`
Hence the point on the x-axis which lies at equal distances from the mentioned points is`(2/5 , 0).`
APPEARS IN
RELATED QUESTIONS
The three vertices of a parallelogram are (3, 4) (3, 8) and (9, 8). Find the fourth vertex.
Find a point on the x-axis which is equidistant from the points (7, 6) and (−3, 4).
Name the quadrilateral formed, if any, by the following points, and given reasons for your answers:
A(4, 5) B(7, 6), C (4, 3), D(1, 2)
Prove that the points (4, 5) (7, 6), (6, 3) (3, 2) are the vertices of a parallelogram. Is it a rectangle.
ABCD is a rectangle whose three vertices are A(4,0), C(4,3) and D(0,3). Find the length of one its diagonal.
Prove that the diagonals of a rectangle ABCD with vertices A(2,-1), B(5,-1) C(5,6) and D(2,6) are equal and bisect each other
Show that A(-4, -7), B(-1, 2), C(8, 5) and D(5, -4) are the vertices of a
rhombus ABCD.
If R (x, y) is a point on the line segment joining the points P (a, b) and Q (b, a), then prove that x + y = a + b.
If the points A(−2, 1), B(a, b) and C(4, −1) ae collinear and a − b = 1, find the values of aand b.
If the points A(1, –2), B(2, 3) C(a, 2) and D(– 4, –3) form a parallelogram, find the value of a and height of the parallelogram taking AB as base.
Write the condition of collinearity of points (x1, y1), (x2, y2) and (x3, y3).
If P (2, p) is the mid-point of the line segment joining the points A (6, −5) and B (−2, 11). find the value of p.
A line segment is of length 10 units. If the coordinates of its one end are (2, −3) and the abscissa of the other end is 10, then its ordinate is
If three points (0, 0), \[\left( 3, \sqrt{3} \right)\] and (3, λ) form an equilateral triangle, then λ =
The distance of the point (4, 7) from the x-axis is
The coordinates of a point on x-axis which lies on the perpendicular bisector of the line segment joining the points (7, 6) and (−3, 4) are
Find the coordinates of point A, where AB is a diameter of the circle with centre (–2, 2) and B is the point with coordinates (3, 4).
The line 3x + y – 9 = 0 divides the line joining the points (1, 3) and (2, 7) internally in the ratio ______.
Assertion (A): The ratio in which the line segment joining (2, -3) and (5, 6) internally divided by x-axis is 1:2.
Reason (R): as formula for the internal division is `((mx_2 + nx_1)/(m + n) , (my_2 + ny_1)/(m + n))`
Distance of the point (6, 5) from the y-axis is ______.
