Advertisements
Advertisements
प्रश्न
AB and AC are the two chords of a circle whose radius is r. If p and q are
the distance of chord AB and CD, from the centre respectively and if
AB = 2AC then proove that 4q2 = p2 + 3r2.
Advertisements
उत्तर
Let AC = a then AB = 2a
seg OM ⊥ chord AC, seg ON ⊥ chord AB.
AM =MC = `a/2` and AN = NB = a
In Δ OMA and Δ ONA, By theorem of Pythagoras,
`AO^2 = AM^2 + MO^2`
`AO^2 = AN^2+ q^2` ....... (1)
`AO^2 = AN^2 + NO^2`
`AO^2 = a^2 + p^2` ........ (2)
From equation (1) and (2)
`(a/2)^2 + q^2 = a^2 + p^2`
`a^2/4 + q^2 = a^2 + p^2`
`a^2 + 4q^2 = 4a^2 + 4p^2`
`4q^2 = 3a^2 + 4p^2`
`4q^2 = p^2 + 3(a^2 + p^2)`
`4q^2 = p^2 + 3r^2 .... ("In" Δ ONA, r^2 = a^2 + p^2)`
APPEARS IN
संबंधित प्रश्न
Show that the points (1, – 1), (5, 2) and (9, 5) are collinear.
Show that four points (0, – 1), (6, 7), (–2, 3) and (8, 3) are the vertices of a rectangle. Also, find its area
Find the distance between the points (0, 0) and (36, 15). Can you now find the distance between the two towns A and B discussed in Section 7.2.
Find the distance of the following points from the origin:
A(5, –12)
Using the distance formula, show that the given points are collinear:
(1, –1), (5, 2) and (9, 5)
Find the distance between the points `A((-8)/5, 2)` and `B(2/5, 2)`.
For what values of k are the points (8, 1), (3, –2k) and (k, –5) collinear ?
Find the distance between the following pairs of point in the coordinate plane :
(7 , -7) and (2 , 5)
Find the distance between the following point :
(sin θ , cos θ) and (cos θ , - sin θ)
Find the coordinates of O, the centre passing through A( -2, -3), B(-1, 0) and C(7, 6). Also, find its radius.
Show that the points (2, 0), (–2, 0), and (0, 2) are the vertices of a triangle. Also, a state with the reason for the type of triangle.
Find the co-ordinates of points on the x-axis which are at a distance of 17 units from the point (11, -8).
A point P lies on the x-axis and another point Q lies on the y-axis.
Write the ordinate of point P.
Given A = (x + 2, -2) and B (11, 6). Find x if AB = 17.
The length of line PQ is 10 units and the co-ordinates of P are (2, -3); calculate the co-ordinates of point Q, if its abscissa is 10.
If the point (x, y) is at equidistant from the point (a + b, b – a) and (a-b, a + b). Prove that ay = bx.
If the distance between point L(x, 7) and point M(1, 15) is 10, then find the value of x.
The coordinates of the point which is equidistant from the three vertices of the ΔAOB as shown in the figure 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 x axis equidistant from I and E is ______.
|
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?
