Advertisements
Advertisements
प्रश्न
Show that the point (11, –2) is equidistant from (4, –3) and (6, 3).
Advertisements
उत्तर
Let P(x1, y1) = P(11, –2), Q(x2, y2) = Q(4, –3), R(x3, y3) = R(6, 3)
By distance formula,
d(P, Q) = `sqrt((x_2 - x_1)^2 + (y_2 - y_1)^2`
= `sqrt((4 - 11)^2 + [-3 - (-2)]^2`
= `sqrt((-7)^2 + (-1)^2`
= `sqrt(49 + 1)`
= `sqrt(50)`
= `5sqrt(2)`
And
d(P, R) = `sqrt((x_3 - x_1)^2 + (y_3 - y_1)^2`
= `sqrt((6 - 11)^2 + [3 - (-2)]^2`
= `sqrt((-5)^2 + (5)^2`
= `sqrt(25 + 25)`
= `sqrt(50)`
= `5sqrt(2)`
Here, d(P, Q) = d(P, R)
∴ Point (11, –2) is equidistant from (4, –3) and (6, 3).
APPEARS IN
संबंधित प्रश्न
Determine if the points (1, 5), (2, 3) and (−2, −11) are collinear.
Name the type of quadrilateral formed, if any, by the following point, and give reasons for your answer:
(4, 5), (7, 6), (4, 3), (1, 2)
An equilateral triangle has two vertices at the points (3, 4) and (−2, 3), find the coordinates of the third vertex.
Show that the quadrilateral whose vertices are (2, −1), (3, 4) (−2, 3) and (−3,−2) is a rhombus.
Find the distance between the points:
A(1, –3) and B(4, –6)
Find the distance between the points `A((-8)/5, 2)` and `B(2/5, 2)`.
Find the distances between the following point.
P(–6, –3), Q(–1, 9)
The perimeter of a triangle with vertices (0, 4), (0, 0) and (3, 0) is ______.
P and Q are two points lying on the x - axis and the y-axis respectively . Find the coordinates of P and Q if the difference between the abscissa of P and the ordinates of Q is 1 and PQ is 5 units.
A line segment of length 10 units has one end at A (-4 , 3). If the ordinate of te othyer end B is 9 , find the abscissa of this end.
Prove that the points (0 , -4) , (6 , 2) , (3 , 5) and (-3 , -1) are the vertices of a rectangle.
PQR is an isosceles triangle . If two of its vertices are P (2 , 0) and Q (2 , 5) , find the coordinates of R if the length of each of the two equal sides is 3.
The distance between the points (3, 1) and (0, x) is 5. Find x.
Find a point on the y-axis which is equidistant from the points (5, 2) and (-4, 3).
The distances of point P (x, y) from the points A (1, - 3) and B (- 2, 2) are in the ratio 2: 3.
Show that: 5x2 + 5y2 - 34x + 70y + 58 = 0.
The points A (3, 0), B (a, -2) and C (4, -1) are the vertices of triangle ABC right angled at vertex A. Find the value of a.
Show that the quadrilateral with vertices (3, 2), (0, 5), (- 3, 2) and (0, -1) is a square.
Find distance between point A(7, 5) and B(2, 5).
Case Study -2
A hockey field is the playing surface for the game of hockey. Historically, the game was played on natural turf (grass) but nowadays it is predominantly played on an artificial turf.
It is rectangular in shape - 100 yards by 60 yards. Goals consist of two upright posts placed equidistant from the centre of the backline, joined at the top by a horizontal crossbar. The inner edges of the posts must be 3.66 metres (4 yards) apart, and the lower edge of the crossbar must be 2.14 metres (7 feet) above the ground.
Each team plays with 11 players on the field during the game including the goalie. Positions you might play include -
- Forward: As shown by players A, B, C and D.
- Midfielders: As shown by players E, F and G.
- Fullbacks: As shown by players H, I and J.
- Goalie: As shown by player K.
Using the picture of a hockey field below, answer the questions that follow:
What are the coordinates of the position of a player Q such that his distance from K is twice his distance from E and K, Q and E are collinear?
If (– 4, 3) and (4, 3) are two vertices of an equilateral triangle, find the coordinates of the third vertex, given that the origin lies in the interior of the triangle.
