Advertisements
Advertisements
प्रश्न
If the point P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b). Prove that bx = ay.
Advertisements
उत्तर
P(x, y) is equidistant from the points A(a + b, b – a) and B(a – b, a + b).
∴ AP = BP
∴ `sqrt([x-(a+b)]^2+[y-(b-a)]^2)=sqrt([x-(a-b)]^2+[y-(a+b)]^2`
∴ [x-(a+b)]2+[y-(b-a)]2 = [x-(a-b)]2+[y-(a+b)]2
∴ x2-2x(a+b)+(a+b)2+y2-2y(b-a)+(b-a)2
= x2-2x(a-b)+(a-b)2+y2-2y(a+b)+(a+b)2
∴ -2x(a+b)-2y(b-a)=-2x(a-b)-2y(a+b)
∴ ax+bx+by-ay=ax-bx+ay+by
∴ 2bx=2ay
∴bx=ay ...(proved)
APPEARS IN
संबंधित प्रश्न
Check whether (5, -2), (6, 4) and (7, -2) are the vertices of an isosceles triangle.
Find the distance between the points
(i) A(9,3) and B(15,11)
Find the distances between the following point.
A(a, 0), B(0, a)
Prove that the points (6 , -1) , (5 , 8) and (1 , 3) are the vertices of an isosceles triangle.
A(2, 5), B(-2, 4) and C(-2, 6) are the vertices of a triangle ABC. Prove that ABC is an isosceles triangle.
The centre of a circle is (2x - 1, 3x + 1). Find x if the circle passes through (-3, -1) and the length of its diameter is 20 unit.
The distances of point P (x, y) from the points A (1, - 3) and B (- 2, 2) are in the ratio 2: 3.
Show that: 5x2 + 5y2 - 34x + 70y + 58 = 0.
The distance between the points A(0, 6) and B(0, -2) 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 ______.
Points A(4, 3), B(6, 4), C(5, –6) and D(–3, 5) are the vertices of a parallelogram.
