Question

The locus of the middle points of the chords of the circle of radius which subtend an angle π/4 at any point on the circumference of the circle is a concentric circle with radius equal to

Solution

Correct option is

Equation of the circle be x2 + y2 = r2. The chord which subtends an angle π/4 at the circumference will subtend a right angle at the centre. Chord joining (r, 0) and (0, r) subtends a right angle at the centre so (h,k) the mid-point of the chord is (r/2, r/2) and locus of (hk) is x2 + y2 =r2/2. 

SIMILAR QUESTIONS

Q1

The locus of a point which moves such that the tangents from it to the two circles x+ y2 – 5x – 3 = 0 and 3x2 + 3y2 + 2x + 4y – 6 = 0 are equal is 

Q2

If the two circles x2 + y2 + 2gx + 2fy = 0 and x2 + y2 + 2g1x + 2f1y = 0 touch each other, then

Q3

If two lines a1x + b1y + c1 = 0 and a2x + b2y + c2 = 0 cut the coordinates axes in concyclic points, then 

Q4

The locus of the point which moves in a plane so that the sum of the squares of its distances from the lines ax + by + c = 0 and

bx – ay + d = 0 is r2, is a circle of radius.  

 

Q5

The locus of the centre of the circle passing through the origin O and the points of intersection of any line through (ab) and the coordinates axis is a

Q6

Four distinct point (1, 0), (0, 1), (0, 0) and (tt) are concyclic for

Q7

If two circles which pass through the points (0, a) and (0, –a) cut each other orthogonally and touch the straight line 

y = mx + c, then

Q8

The coordinates of two point P and Q are (2, 3) and (3, 2) respectively. Circles are described on OP and OQ as diameters; O being the origin, then length of the common chord is

Q9

The circle x2 + y2 – 6x – 4y + 9 = 0 bisects the circumference of the circle x2 + y2 – (λ + 4)x – (λ + 2)y + (5λ + 3) = 0 if λ is equal to

Q10

The lengths of the tangents from two points A and B to a circle are l and l’ respectively. If the points are conjugate with respect to the circle, then (AB)2 is equal to