If the circles x^{2}+y^{2}+2ax+cy+a=0\; and\; x^{2}+y^{2}-3ax+dy-1=0  intersect in two distinct points P\; and\; Q then the line 5x+by-a=0  passes through P\; and\; Q for

  • Option 1)

    no value of a

  • Option 2)

    exactly one value of a

  • Option 3)

    exactly two value of a

  • Option 4)

    infinitely many value of a

 

Answers (1)

As we learnt in 

General form of a circle -

x^{2}+y^{2}+2gx+2fy+c= 0
 

- wherein

centre = \left ( -g,-f \right )

radius = \sqrt{g^{2}+f^{2}-c}

 

 x^{2}+y^{2}+2ax+cy+a=0

x^{2}+y^{2}-3ax+dy-1=0

Subtracting , we get

5ax+\left ( c-d \right )y+\left ( a+1 \right )=0

Also 5x+by-a=0

\frac{5a}{5}=\frac{c-d}{b}=-\left ( \frac{a+1}{a} \right )

a^{2}=-a-1

=> a^{2}+a+1=0

Which has no real root.

 


Option 1)

no value of a

Correct option

Option 2)

exactly one value of a

Incorrect option    

Option 3)

exactly two value of a

Incorrect option    

Option 4)

infinitely many value of a

Incorrect option    

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