Multiple choice

The equation of two circles are $x^{2}+y^{2}+2\lambda x+5=0$ and $x^{2}+y^{2}+2\lambda y+5=0$. $P$ is any point on the line $x-y=0$. If $PA$ and $PB$ are the lengths of the tangents from $P$ to the two circles and $PA=3$ then $PB$ is equal to

  1. $1.5$
  2. $6$
  3. $3$

  4. none of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The length of the tangent from a point to a circle is given by sqrt(S1). For both circles, the power of point P(h, h) is h^2 + h^2 + 2*lambda*h + 5. Since the power is the same for both circles, the tangent lengths PA and PB must be equal. Given PA = 3, PB must also be 3.

AI explanation

Any point P on the line x - y = 0 has identical coordinates, meaning its distances to the centres of both circles and its relative position to the constants are identical. The power of point P, which determines the square of the tangent length, evaluates to the exact same value for both equations because substituting y for x makes the equations equal. Since PA = 3, PB must also equal 3.

Keep practicing — related questions