Question

The line x + y = a, meets the axis of x and y at A and B respectively. A triangle AMN is inscribed in the triangle OABO being the origin, with right angle at NM and N lie respectively on OB and AB. If the area of the triangle AMN is 3/8 of the area of the triangle OAB, then AN/BN is equal to.

Solution

Correct option is

3

 

     

                                   

When (a, 0) and (0, a) are the coordinates of A and B respectively.

Now equation of MN perpendicular to AB is 

                             

 

     

So the coordinates of M are     

Therefore, area of the triangle AMN is

                           

Also area of the triangle OAB = a2/2.

So that according to the given condition.

                        

    

   

For λ = 1/3, M lies outside the segment OB and hence the required value of λ is 3.

 

SIMILAR QUESTIONS

Q1

The line L has intercepts a and b on the coordinate axes. The coordinate axes are rotated through a fixed angle, keeping the origin fixed. If p andq are the intercepts of the line L on the new axes, then 

                                                          

Q2

If P is a point (xy) on the line, y = –3x such that P and the point (3, 4) are on the opposite sides of the line 3x – 4y = 8, then   

 

Q3

The area enclosed by 2|x| + 3|y≤ 6 is

Q4

Let O be the origin, A (1, 0) and B (0, 1) and P (xy) are points such thatxy > 0 and x + y < 1, then

Q5

If a line joining two points A (2, 0) and B (3, 1) is rotated about A in anticlockwise direction through and angle 15o, then equation of the line is the new position is

Q6

An equation of a line through the point (1, 2) whose distance from the point (3, 1) has the greatest value is

Q7

Let 0 < α < π/2 be a fixed angle. If P = (cos θ, sin θ) and Q = (cos (α – θ), sin (α – θ) then Q is obtained from P by

Q8

On the portion of the straight line x + y = 2 which is intercepted between the axes, a square is constructed, away from the origin, with this portion as one of its side. If p denotes the perpendicular distance of a side of this square from the origin, then the maximum value of p is 

Q9

The line x + y = 1 meets x-axis at A and y-axis at BP is the mid-point ofAB (Fig). P is the foot of the perpendicular from P to OAM1 is that from P1 to OPP2 is that from M1 to OAM2 is that from P2 to OPP3is that from M2 to OA and so on. If Pn denotes the nth foot of the perpendicular on OA from Mn – 1, then OPn =  

Q10

The point (4, 1) undergoes the following transformation successively.

(i) Reflection about the line y = x

(ii) Translation through a distance 2 units along the positive direction of x-axis.

(iii) Rotation through an anlge π/4 about the origin in the anticlockwise direction  

(iv) Reflection about x = 0

The final position of the given point is