If The Two Lines  cut The Co-ordinate Axes In Concyclic Points, Prove That a1a2 = b1b2 and Find The Equation Of The Circle.

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Question

If the two lines  cut the co-ordinate axes in concyclic points, prove that a1a2 = b1b2 and find the equation of the circle.

Solution

Correct option is

 

The equation of any curve passing through 

                  

                  

                    

Where λ is a parameter.

This curve will represent a circle. If the coefficient of x2 = coefficient of y2,  

  

and if the coefficient of xy = 0  

then                 a1b2 + a2b1 + λ = 0 

 

Substituting the value of λ in (1) then

From (2), b1b2 = a1a2  

∴ Equation of required circle is

   

SIMILAR QUESTIONS

Q1

The circle x2 + y2 – 4x – 8y + 16 = 0 rolls up the tangent to it at  by 2 units, assuming the x-axis as horizontal, find the equation of the circle in the new position.  

Q2

 

Find the equation of the circle of minimum radius which contains the three circles 

                   x2 – y2 – 4y – 5 = 0 

               x2 + y2 + 12x + 4y + 31 = 0  

and         x2 + y2 + 6x + 12y + 36 = 0

Q3

Find the equation of the circle passing through (1, 0) and (0, 1) and having the smallest possible radius.

Q4

 

If the line x cos α + y sin α = p cuts the circle x2 + y2 = a2 in M and N, then show that the circle, whose diameter is MN, is 

              

Q5

A line meets the co-ordinate axes at A and B. A circle is circumscribed about the triangle OAB. If the distance of the points A and B from the tangent at O, the origin, to the circle are m and n respectively, find the equation of the circle.

Q6

Tangents PQPR are drawn to the circle x2 + y2 = 36 from the point P(–8, 2) touching the circle at QR respectively. Find the equation of the circumcircle of the âˆ†PQR.

Q7

Let 1 and C2 be two circles with C2 lying inside C1. A circle C lying inside C1 touches C1 internally and C2 externally. Identify the locus of the centre of C.

Q8

The circle x2 + y2 – 4x – 4y + 4 = 0 is inscribed in a triangle which has two of its sides along the co-ordinate axes. The locus of the circumcentre of the triangle is x + y – xy + k(x2 + y2)1/2 = 0. Find k.

Q9

Tangents TP and TQ are drawn from a point T to the circle x2 + y2 = a2. If the point T lies on the line px + qy = r, find the locus of centre of the circumcircle of triangle TPQ.