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Question
In what ratio does the point P(−4, y) divides the line segment joining the points A(−6, 10) and B(3, −8)? Hence find the value of y.
Solution
Let P divides the line segment AB in the ratio k : 1
Using section formula
x = `(m_1x_2 + m_2x_1)/(m_1+m_2), y = (m_1y_2 + m_2y_1)/(m_1+m_2)`
A(-6, 10) and B(3, 8)
m1 : m2 = k : 1
plugging values in the formula we get
- 4 = `( k xx 3 + 1 xx (-6))/(k + 1), y = ( k xx (- 8) + 1 xx 10)/(k + 1)`
- 4 = `( 3k - 6)/(k + 1), y = (-8k + 10)/(k + 1)`
Considering only x coordinate to find the value of k
- 4k - 4 = 3k - 6
- 7k = - 2
k = `2/7`
k : 1 = 2 : 7
Now, we have to find the value of y
so, we will use section formula only in y coordinate to find the value of y.
y = `(2 xx (- 8) + 7 xx 10)/(2 + 7)`
y = `( - 16 + 70 )/(9)`
y = 6
Therefore, P divides the line segment AB in 2 : 7 ratio
And value of y is 6.
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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 x axis equidistant from I and E is ______.
A circle has its centre at the origin and a point P(5, 0) lies on it. The point Q(6, 8) lies outside the circle.
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.
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Case Study Trigonometry in the form of triangulation forms the basis of navigation, whether it is by land, sea or air. GPS a radio navigation system helps to locate our position on earth with the help of satellites. |
- Make a labelled figure on the basis of the given information and calculate the distance of the boat from the foot of the observation tower.
- After 10 minutes, the guard observed that the boat was approaching the tower and its distance from tower is reduced by 240(`sqrt(3)` - 1) m. He immediately raised the alarm. What was the new angle of depression of the boat from the top of the observation tower?
