Q. 44.0( 3 Votes )
If the abscissae and ordinates of two points P and Q are roots of the equations x2 + 2ax – b2 = 0 and x2 + 2px – q2 = 0 respectively, then write the equation of the circle with PQ as diameter.
Answer :
Given that points, we need to find the equation of the circle whose ends of the diameter are P and Q.
It is also told that the abscissae of two points P and Q are the roots of x2 + 2ax - b2 = 0 and ordinates are the roots of x2 + 2px - q2 = 0.
Let us first find the roots of each quadratic equation.
For x2 + 2ax - b2 = 0
⇒ 
⇒ 
⇒ 
..... (1)
For x2 + 2px - q2 = 0
⇒ 
⇒ 
⇒ 
..... (2)
From (1) and (2) we get,
⇒ 
⇒ 
We know that the centre is the mid - point of the diameter.
⇒ Centre(C) = 
⇒ C = (- a, - p)
We have a circle with centre (- a, - p) and passing through the point 
.
We know that the radius of the circle is the distance between the centre and any point on the radius. So, we find the radius of the circle.
We know that the distance between the two points (x1,y1) and (x2,y2) is 
.
Let us assume r is the radius of the circle.
⇒ 
⇒ 
⇒ 
We know that the equation of the circle with centre (p, q) and having radius ‘r’ is given by:
⇒ (x - p)2 + (y - q)2 = r2
Now we substitute the corresponding values in the equation:
⇒ 
⇒ x2 + 2ax + a2 + y2 + 2py + p2 = a2 + b2 + p2 + q2
⇒ x2 + y2 + 2ax + 2py - b2 - q2 = 0
∴The equation of the circle is x2 + y2 + 2ax + 2py - b2 - q2 = 0.
Rate this question :
Lecture on radical axis of 2 circles48 mins
Lecture on Common tangents & angle of intersection of 2 circles56 mins
Lecture on Tangents to a Circle57 mins
General & parametric form of circle56 mins
Quiz on Tangents to a Circle48 mins
Interactive Quiz on radical axis of 2 circles56 mins
Interactive Quiz on Locus involving circles42 mins
Family of circles55 mins
Practice Problems on Equation of Circle54 mins
Interactive quiz on equation of circle50 mins
Find the equation of the circle circumscribing the rectangle whose sides are x – 3y = 4, 3x + y = 22, x – 3y = 14 and 3x + y = 62.
RD Sharma - MathematicsFind the equation of the circle
Which touches both the axes and passes through the point (2, 1)
RD Sharma - MathematicsFind the equation of the circle
Which touches x - axis at a distance of 5 from the origin and radius 6 units.
RD Sharma - MathematicsIf the lines 2x – 3y = 5 and 3x – 4y = 7 are the diameters of a circle of area 154 squares units, then obtain the equation of the circle.
RD Sharma - MathematicsThe sides of a squares are x = 6, x = 9, y = 3 and y = 6. Find the equation of a circle drawn on the diagonal of the square as its diameter.
RD Sharma - MathematicsThe line 2x – y + 6 = 0 meets the circle x2 + y2 – 2y – 9 = 0 at A and B. Find the equation of the circle on AB as diameter.
RD Sharma - MathematicsABCD is a square whose side is a; taking AB and AD as axes, prove that the equation of the circle circumscribing the square is x2 + y2 – a(x + y) = 0.
RD Sharma - MathematicsThe circle x2 + y2 – 2x – 2y + 1 = 0 is rolled along the positive direction of x - axis and makes one complete roll. Find its equation in new - position.
RD Sharma - Mathematics