Problem 17 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the circle \(x^{2}+y^{2}+2 g x+2 f y+c=0\) bisects the circumference of the circles \(x^{2}+y^{2}+2 g^{\prime} x+2 f^{\prime} y+c^{\prime}=0\), then (A) \(2 g^{\prime}\left(g-g^{\prime}\right)+2 f^{\prime}\left(f-f^{\prime}\right)=c-c^{\prime}\) (B) \(g^{\prime}\left(g-g^{\prime}\right)+f^{\prime}\left(f-f^{\prime}\right)+c-c^{\prime}=0\) (C) \(2 g\left(g-g^{\prime}\right)+2 f\left(f-f^{\prime}\right)=c-c^{\prime}\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (D).

Step 1: Identify condition for bisecting circle

For the given circles \(x^2 + y^2 + 2gx + 2fy + c = 0\) and \(x^2 + y^2 + 2g'x + 2f'y + c' = 0\), the first circle bisects the circumference of the second circle if they intersect orthogonally. This means the gradients of the tangents at the points of intersection are perpendicular.

Step 2: Use the condition for orthogonal circles

The condition for two circles to be orthogonal is \(2gg' + 2ff' = c + c'\). This arises because, at the intersection points, the product of the slopes of the tangents is \(-1\), reflecting perpendicularity.

Step 3: Analyze given answers

Given options must be evaluated considering the orthogonal condition derived in Step 2. None of the given options \( (A), (B), (C) \) directly match the derived equation \( 2gg' + 2ff' = c + c' \).

Step 4: Conclude

Since options \( (A), (B), (C) \) do not satisfy the equation necessary for orthogonality (and hence bisecting each other), the correct answer is \( (D) \) 'none of these'.

Key Concepts

Orthogonal Circles

Orthogonal circles are a pair of circles that intersect at right angles. This is a very special type of intersection where the tangents to the circles at the points of intersection are perpendicular to each other. The concept of orthogonality is crucial for understanding various geometric properties of intersecting circles.

When two circles intersect orthogonally, you can use this condition to verify certain geometric properties or solve problems involving these circles. For orthogonal circles, the equation that describes their relationship is:

• \( 2gg' + 2ff' = c + c' \)

Here, \(g, f, \) and \(c\) are parameters from the general equations of the circles. Essentially, these parameters describe the position and size of the circles.

This formula arises because, at the intersection points, the two circles exert a perpendicular force on each other's tangents. Hence, the condition equates to a geometric requirement that the slopes of these tangents multiply to

• -1

indicating perpendicular directions.

Intersection of Circles

The intersection of circles is an interesting concept in geometry where two circles converge at one or more points. In the simplest case, two circles can intersect at two points which lie on both circles' circumferences. These points of intersection carry rich geometric implications and can be utilized to solve various mathematical problems.

To find the locus of intersection points analytically, one needs to solve the simultaneous equations of the circles. For the circles given in the equation:

• First Circle: \(x^2 + y^2 + 2gx + 2fy + c = 0\)
• Second Circle: \(x^2 + y^2 + 2g'x + 2f'y + c' = 0\)

By subtracting these equations, we can derive a new equation in \(x\) and \(y\) that represents the set of intersection points.

Understanding intersections aids in solving geometric challenges such as determining if circles bisect or viewing points of tangency, useful in more complex constructions.

Circle Bisecting Circumference

When we say a circle bisects the circumference of another circle, it means that the first circle divides the circumference of the second into two equal arcs. For this to happen geometrically, the circles often intersect at points satisfying specific conditions such as orthogonality.

In the context of this problem, if one circle bisects another's circumference, it implies there are specific positions and sizes that allow this equal division. The primary condition explored here is that the bisecting circle must intersect orthogonally with the circle it is bisecting.

• Condition: The formula \( 2gg' + 2ff' = c + c' \) needs to hold true.

This ensures that the first circle intersects the second circle at right angles, essentially slicing the second circle's circumference into equal halves.

Analyzing the equation of a circle, it becomes easier to predict possible intersections and bisecting behavior, supporting advanced geometric explorations and solutions.