Question

Find the equation of the circle which passes through the points (4, 1), (6, 5) and has its centre on the line 4x + y = 16.

Solution

Correct option is

x2 + y2 – 6x – 8y + 15 = 0

 

Let the equation of circle be

                 x2 + y2 + 2gx + 2fy + c = 0                    …(1)

since the centre of (1) i.e., (–g, f) lies on 4x + y = 16

then              –4g – f = 16

or                   4g + f + 16 = 0                …(2)

Since the points (4, 1) and (6, 5) lie on the circle x2 + y2 + 2gx + 2fy + c= 0, we get the equations

         16 + 1 + 8g + 2f + c + 0    or   17 + 8g + 2f + c = 0     …(3)

and  36 + 25 + 12+ 10 f + c = 0 or 61 + 12g + 10f + c = 0 ...(4)

subtracting (3) and (4), then  

                                   44 + 4g + 8f = 0                      …(5)

Solving (2) and (5), we get

                        f = – 4 and g = – 3  

Now from (3),     

                       17 – 24 – 8 + c = 0

⇒                    c = 15

Hence, the equation of circle becomes

            x2 + y2 – 6x – 8y + 15 = 0

SIMILAR QUESTIONS

Q1

Find the equation of the circle concentric with the circle x2 + y2 – 8x + 6y– 5 = 0 and passing through the point (–2, –7).

Q2

A circle has radius 3 units and its centre lies on the line y = x – 1. Find the equation of the circle if it passes through (7, 3).

Q3

 

Find the area of an equilateral triangle inscribed in the circle

                       x2 + y2 + 2gx + 2fy + c = 0     

Q4

 

Find the parametric form of the equation of the circle

                               x2 + y2 + px + py = 0

Q5

 

If the parametric of form of a circle is given by

(i) x = – 4 + 5 cos θ and y = – 3 + 5 sin θ  

(ii) x = a cos α + b sin α and y = a sin α – b cos α

Find its Cartesian form.

Q6

Find the equation if the circle the end points of whose diameter are the centres of the circle x2 + y2 + 6x – 14y = 1 and x2 + y2 – 4x + 10y = 2.

Q7

The sides of a square are x = 2, x = 3, y = 1and y = 2. Find the equation of the circle drawn on the diagonals of the square as its diameter.

Q8

Find the equation of the circum circle of the quadrilateral formed by the four lines ax + by ± c = 0 and bx – ay ± c = 0.

Q9

The abscissa of two points A and B are the roots of the equation x2 + 2ax – b2 = 0 and their ordinates are the roots of the equation x2 + 2px –q2 = 0. Find the equation and the radius of the circle with AB as diameter.

Q10

Find the equation of the circle passing through the three non-collinear points (1, 1), (2, –1) and (3, 2).