Advertisements
Advertisements
प्रश्न
Find the value of a when the distance between the points (3, a) and (4, 1) is `sqrt10`
उत्तर
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)`
The distance between two points (3, a) and (4, 1) is given as `sqrt10`. Substituting these values in the formula for distance between two points we have
`sqrt10 = sqrt((3 - 4)^2 `+ (a - 1)^2)`
`sqrt10 = sqrt((-1)^2 + (a - 1))`
Now, squaring the above equation on both sides of the equals sign
`10 = (-1)^2 + (a - 1)^2`
`10 = 1 + (a^2 + 1 - 2a)`
`8 = a^2 - 2a`
Thus we arrive at a quadratic equation. Let us solve this now,
`a^2 - 2a - 8 = 0`
`a^2 -4a + 2a - 8 = 0`
a(a - 4) + 2(a - 4) = 0
(a - 4)(a + 2) = 0
The roots of the above quadratic equation are thus 4 and −2.
Thus the value of ‘a’ could either be 4 or -2
APPEARS IN
संबंधित प्रश्न
Find the values of y for which the distance between the points P (2, -3) and Q (10, y) is 10 units.
Find value of x for which the distance between the points P(x,4) and Q(9,10) is 10 units.
Find the distance of a point (12 , 5) from another point on the line x = 0 whose ordinate is 9.
Find the coordinate of O , the centre of a circle passing through A (8 , 12) , B (11 , 3), and C (0 , 14). Also , find its radius.
Prove taht the points (-2 , 1) , (-1 , 4) and (0 , 3) are the vertices of a right - angled triangle.
Prove that the points (0 , -4) , (6 , 2) , (3 , 5) and (-3 , -1) are the vertices of a rectangle.
Show that the points P (0, 5), Q (5, 10) and R (6, 3) are the vertices of an isosceles triangle.
AOBC is a rectangle whose three vertices are A(0, 3), O(0, 0) and B(5, 0). The length of its diagonal is ______.
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:
The point on y axis equidistant from B and C is ______.
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.
