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                …(i)

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

Then        –4g – f = 16

Or              4g + f + 16 = 0                  …(ii)

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

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

And 36 + 25 + 12g + 10f + c = 0 or 61 + 12g + 10f + c = 0   …(iv)

Subtracting (3) from (4), then

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

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 the end points of whose diameter are the centers of the circle.

                  x2 + y2 + 6x – 14y = 1 and x2 + y2 – 4x + 10y = 2  

Q2

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

Q3

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

Q4

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

Q5

Find the equation of the circle whose diameter is the line joining the points (–4, 3) and (12, –1). Find also the intercept made by it on y axis.

Q6

 

Find the equation of circle which touches axis of y at a distance 4 units from the origin and cuts the intercept of 6 units from the axis of x.

Equation of circle in intercepts  

               

                

Q7

Find the equation of the circle which passes through the origin and makes intercepts of length a and b on axis of x and y respectively.

Q8

A circle of radius 2 lies in the first quadrant and touches both the axes of coordinate. Find the equation of circle with centre at (6, 5) and touching the above circle externally.

Q9

A circle of radius 5 units touches the coordinates axes in 1st quadrant. If the circle makes one complete roll on x axis along positive direction of xaxis. Find the equation in new position. 

Q10

 

Discus the position of the points (1, 2) and (6, 0) with respect to the circle.

                 x2 + y2 – 4x + 2y – 11 = 0