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.
Advertisements
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
If a ≠ b ≠ 0, prove that the points (a, a2), (b, b2) (0, 0) will not be collinear.
Find the co-ordinates of points of trisection of the line segment joining the point (6, –9) and the origin.
Find the values of x for which the distance between the points P(x, 4) and Q(9, 10) is 10 units.
Find the distance between the following pair of point.
T(–3, 6), R(9, –10)
Find the relation between x and y if the point M (x,y) is equidistant from R (0,9) and T (14 , 11).
Prove that the following set of point is collinear :
(5 , 5),(3 , 4),(-7 , -1)
Find the distance between the following pair of points:
`(sqrt(3)+1,1)` and `(0, sqrt(3))`
Find the distance between the origin and the point:
(-5, -12)
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 (2, -1) is equidistant from the points (a, 7) and (-3, a). Find a.
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.
Given A = (3, 1) and B = (0, y - 1). Find y if AB = 5.
Calculate the distance between the points P (2, 2) and Q (5, 4) correct to three significant figures.
Find the point on y-axis whose distances from the points A (6, 7) and B (4, -3) are in the ratio 1: 2.
Find the distance of the following points from origin.
(a+b, a-b)
Find distance between point A(7, 5) and B(2, 5).
The point which divides the lines segment joining the points (7, -6) and (3, 4) in ratio 1 : 2 internally lies in the ______.
A circle drawn with origin as the centre passes through `(13/2, 0)`. The point which does not lie in the interior of the circle 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?
Show that points A(–1, –1), B(0, 1), C(1, 3) are collinear.
