Advertisements
Advertisements
Question
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.
Solution
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
RELATED QUESTIONS
Find the distance between the points
A(-6,-4) and B(9,-12)
If P (x , y ) is equidistant from the points A (7,1) and B (3,5) find the relation between x and y
Find the distance between the following pair of points.
R(0, -3), S(0, `5/2`)
Find the distance between the following pair of point in the coordinate plane.
(1 , 3) and (3 , 9)
Find the distance between the following pairs of point in the coordinate plane :
(7 , -7) and (2 , 5)
Find the distance of the following point from the origin :
(5 , 12)
Find the distance of the following point from the origin :
(8 , 15)
Find the distance of a point (12 , 5) from another point on the line x = 0 whose ordinate is 9.
Prove that the following set of point is collinear :
(5 , 5),(3 , 4),(-7 , -1)
Prove that the following set of point is collinear :
(4, -5),(1 , 1),(-2 , 7)
Prove that the points (5 , 3) , (1 , 2), (2 , -2) and (6 ,-1) are the vertices of a square.
Find the distance between the following pairs of points:
`(3/5,2) and (-(1)/(5),1(2)/(5))`
A point P lies on the x-axis and another point Q lies on the y-axis.
If the abscissa of point P is -12 and the ordinate of point Q is -16; calculate the length of line segment PQ.
The distance between points P(–1, 1) and Q(5, –7) is ______
Find distance between point A(– 3, 4) and origin O
Using distance formula decide whether the points (4, 3), (5, 1), and (1, 9) are collinear or not.
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 ______.
The points (– 4, 0), (4, 0), (0, 3) are the vertices of a ______.
Read the following passage:
|
Use of mobile screen for long hours makes your eye sight weak and give you headaches. Children who are addicted to play "PUBG" can get easily stressed out. To raise social awareness about ill effects of playing PUBG, a school decided to start 'BAN PUBG' campaign, in which students are asked to prepare campaign board in the shape of a rectangle: One such campaign board made by class X student of the school is shown in the figure.
|
Based on the above information, answer the following questions:
- Find the coordinates of the point of intersection of diagonals AC and BD.
- Find the length of the diagonal AC.
-
- Find the area of the campaign Board ABCD.
OR - Find the ratio of the length of side AB to the length of the diagonal AC.
- Find the area of the campaign Board ABCD.
A point (x, y) is at a distance of 5 units from the origin. How many such points lie in the third quadrant?
