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Radical Axis

If the circle...

Question

If the circles $x^{2} + y^{2} + 2 a x + c y + a = 0$ and $x^{2} + y^{2} - 3 a x + d y - 1 = 0$ interesect in two distinct points P and Q then the line 5x + by - a = 0 passes through P and Q for
(2005)

exactly for one value of a

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no value of a

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infinitely many values of a

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exactly two values of a

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Solution

The correct option is B
no value of a

$S_{1} = x^{2} + y^{2} + 2 a x + c y + a = 0$
$S_{2} = x^{2} + y^{2} - 3 a x + d y - 1 = 0$
Equation of radical axis of $S_{1}$ and $S_{2}$
$S_{1} - S_{2} = 0 \Rightarrow 5 a x + (c - d) y + a + 1 = 0$
Given that $5 x + b y - a = 0$ passes through P and Q

$\Rightarrow \frac{a}{1} = \frac{c - d}{b} = \frac{a + 1}{- a} \Rightarrow a + 1 = - a^{2}$
$\Rightarrow a^{2} + a + 1 = 0 \Rightarrow a \notin R .$
Not real value of a.

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Similar questions

If the circles $x^{2} + y^{2} + 2 a x + c y + a = 0$ and $x^{2} + y^{2} - 3 a x + d y - 1 = 0$ interesect in two distinct points P and Q then the line 5x + by - a = 0 passes through P and Q for
(2005)

Q. If the circles

x^{2} + y^{2} + 2 a x + c y + a = 0

and

x^{2} + y^{2} - 3 a x + d y - 1 = 0

intersect in two distinct points

P

and

Q

, then the line

5 x + b y - a = 0

passes through

P

and

Q

for

Q. lf the circles

x^{2} + y^{2} + 2 a x +

+ a = 0

and

x^{2} + y^{2} - 3 a x + d y - 1 = 0

intersect in two distinct points

P

and

Q

then the line

5 x + b y - a = 0

passes through

P

and

Q

for

If the circles $x^{2} + y^{2} + 2 a x + c y + a = 0$ and $x^{2} + y^{2} - 3 a x + d y - 1 = 0$ interesect in two distinct points P and Q then the line 5x + by - a = 0 passes through P and Q for
(2005)

Q. The two circles

x^{2} + y^{2} + 2 a x - c^{2} = 0

and

x^{2} + y^{2} + 2 a b x - c^{2} = 0

meet in points P and Q. Parallel lines are drawn through P and Q meet the circles in point R and S. Prove that the locus of mid-point of RS is the circle.

x^{2} + y^{2} + (a + b) x = 0

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If the circles x2+y2+2ax+cy+a=0 and x2+y2−3ax+dy−1=0 interesect in two distinct points P and Q then the line 5x + by - a = 0 passes through P and Q for (2005)

The correct option is B no value of a S1=x2+y2+2ax+cy+a=0 S2=x2+y2−3ax+dy−1=0 Equation of radical axis of S1 and S2 S1−S2=0⇒5ax+(c−d)y+a+1=0 Given that 5x+by−a=0 passes through P and Q ⇒a1=c−db=a+1−a⇒a+1=−a2 ⇒a2+a+1=0 ⇒a∉R. Not real value of a.

If the circles $x^{2} + y^{2} + 2 a x + c y + a = 0$ and $x^{2} + y^{2} - 3 a x + d y - 1 = 0$ interesect in two distinct points P and Q then the line 5x + by - a = 0 passes through P and Q for
(2005)