Is That Lambda Of Any Use?

Geometry Level 4

Suppose P P is a point on the line x − y = 0 x-y=0 . P A PA and P B PB are the lengths of tangents drawn from P P to the circles whose equations are x 2 + y 2 + 2 λ x + 5 = 0 x^2+y^2+2\lambda x + 5=0 and x 2 + y 2 + 2 λ y + 5 = 0 x^2+y^2+2\lambda y + 5=0 respectively, such that P A = 3 PA=3 . Evaluate P B PB to correct two decimal places.


The answer is 3.00.

Nihar Mahajan by Nihar Mahajan , 20, India

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3 solutions

Line x-y=0 is the radical axis of the two circles.Since P is a point on the radical axis of the two circles,we must have PA=PB.Therefore,PB=3.

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Ameya Daigavane
Jun 6, 2016

By definition, the radical axis of two circles contains the points of equal power with respect to both circles.

The power of a point outside a circle can be shown to be equal to the square of the length of the tangent to the circle.

Hence, P A 2 = P B 2 ⇒ P A = P B PA^2 = PB^2 \Rightarrow PA = PB

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Since y=x is the line of symmetry for the full configuration, the length of the tangent must be same. If the figure is folded along y=x, the to tangent will coincide. Or one is the reflection of the other along the y=x.

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A circle is not a function ( by definition of function).So,your statement that the two circles are inverse functions of each other is incorrect.

Indraneel Mukhopadhyaya - 5 years ago

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I agree. Thanks. I am removing the statement.

Niranjan Khanderia - 5 years ago

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