Question

An equation of the chord of the circle x2 + y2 = a2 passing through the point (2, 3) farthest from the centre is

Solution

Correct option is

2x + 3y = 13

 

Let P(2, 3) be the given point, M be the middle point of a chord of the circle x2 + y2 = a2 through P. Then the distance of the centre O of the circle from the chord is OM.  

and                  (OM)2 = (OP)2 – (PM)2                    (Fig.)

which is maximum when PM is minimum, i.e. M coincides with P i.e. Pis the middle point of the chord.  

Hence the equation of the chord is   

2.x + 3.y – a2 = (2)2 + (3)2 – a2    ⇒   2x + 3y = 13.

                                                                                        

SIMILAR QUESTIONS

Q1

An equilateral triangle is inscribed in the circle x2 + y2 = a2 with the vertex at (a, 0). The equation of the side opposite to this vertex is

Q2

The lines 2x – 3y = 5 and 3x – 4y = 7 are the diameters of a circle of area 154 square units. An equation of this circle is (π = 22/7)

Q3

The equation f a circle with origin as centre and passing through the vertices of an equilateral triangle whose median is of length 3a is

Q4

A line is drawn through the point P(3, 11) to cut the circle x2 + y2 = 9 at A and B. Then PAPB is equal to

Q5

Two rods of lengths a and b slide along the x-axid and y-axis respectively in such a manner that their ends are concyclic. The locus of the centre of the circle passing through the end points is

Q6

If the point (1, 4) lies inside the circle x2 + y2 – 6x – 10y + p = 0 and the circle does not touch or interest the coordinates axes, then

Q7

If the line x cos α + y sin α = p represents the common chord APQB of the circle x2 + y2 = a2 and x2 + y2 = b2 (a > b) as shown in the Fig, then AP is equal to

Q8

Two points P and Q are taken on the line joining the points A (0, 0) and B (3a, 0) such that AP = PQ = QB. Circles are drawn onAPPQ, and QB as diameters. The locus of the point S, the sum of the squares of the lengths of the tangents from which to the three circles is equal to b2, is

Q9

If OA and OB are two equal chords of the circle x2 + y2 – 2x + 4y = 0 perpendicular to each other and passing through the origin O, the slopes of OA and OB are the roots of the equation 

Q10

The lengths of the intercepts made by any circle on the coordinates axes are equal if the centre lies on the line (s) represented by