Problem 59 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The equation of the system of coaxal circles that are tangent at \((\sqrt{2}, 4)\) to the locus of the point of intersection of mutually \(\perp\) tangents to the circle \(x^{2}+y^{2}=9\), is (A) \(\left(x^{2}+y^{2}-18\right)+\lambda(\sqrt{2} x+4 y-18)=0\) (B) \(\left(x^{2}+y^{2}-18\right)+\lambda(4 x+\sqrt{2 y}-18)=0\) (C) \(\left(x^{2}+y^{2}-16\right)+\lambda(\sqrt{2 x}+4 y-16)=0\) (D) none of these

Step-by-Step Solution

Answer

(A) \((x^2 + y^2 - 18) + \lambda(\sqrt{2} x + 4 y - 18) = 0\)."

Step 1: Understand the given circle

The given circle equation is \(x^2 + y^2 = 9\), which represents a circle with center \((0,0)\) and radius \(3\).

Step 2: Determine the locus of intersection of perpendicular tangents

The locus of the intersection of mutually perpendicular tangents to a circle is a circle with the same center and twice the radius. Thus, the locus is \(x^2 + y^2 = 18\).

Step 3: Equation of coaxal circles

Coaxial circles with a fixed circle \(x^2 + y^2 = 18\) have the equation \((x^2 + y^2 - 18) + \lambda(L) = 0\), where \(L\) is a line.

Step 4: Line condition for tangency at a point

For the circles to be tangent at \((\sqrt{2}, 4)\), substituting this point into \(L\) gives the line equation: \(\sqrt{2}x + 4y = 18\).

Step 5: Construct the final equation

Using the form from Step 3 and the line \(L = \sqrt{2}x + 4y - 18\), the equation becomes \((x^2 + y^2 - 18) + \lambda(\sqrt{2} x + 4 y - 18) = 0\).

Key Concepts

Locus of Intersection

The concept of a locus refers to a set of points that satisfy a certain condition. In the context of circles, the locus of the intersection of mutually perpendicular tangents is particularly interesting. Given a circle with equation \(x^2 + y^2 = 9\), its center is at the origin \((0, 0)\) and it has a radius of 3. When exploring the locus of points formed by the intersection of tangents that meet at right angles, it helps to remember that these points create a new circle. This circle shares the same center as the original circle but its radius is doubled. Thus, the locus of intersection for tangents perpendicular to a circle with radius 3 is a larger circle with radius 6. Therefore, the equation becomes \(x^2 + y^2 = 18\).
This transformation reveals how tangents can define a new geometric space, capturing not only the presence of the original circle but also extending beyond it.

Equation of Circle

The equation of a circle is a fundamental concept in geometry and is essential for understanding circles' properties and relationships. Typically, a circle centered at \((h, k)\) with radius \(r\) is represented by \((x - h)^2 + (y - k)^2 = r^2\). In simpler terms, every point \((x, y)\) on the circle is at a distance \(r\) from the center.
In our exercise, we have the circle \(x^2 + y^2 = 18\), indicating a center at \((0,0)\) with a radius of 6. Coaxial circles, like those described in the problem, share a common axis—in this case, they share the same center. Their general equation takes the form \((x^2 + y^2 - 18) + \lambda L = 0\), where \(L\) is a linear element.

Tangent to a Circle

A tangent to a circle is a line that just "touches" the circle at exactly one point, providing a way to explore and define further geometric properties. In our context, when the problem specifies tangency at \((\sqrt{2}, 4)\), it dictates specific conditions for possible lines.

• These lines do not penetrate the circle, ensuring only one point of contact.
• They create a perpendicular radius at the point of contact, visible in special scenarios such as mutually perpendicular tangents.

For this question, finding the tangent's equation involves substituting said point into the linear form \(L = \sqrt{2}x + 4y - 18\). This condition guarantees tangent touching, simplifying further circle construction.

Mutually Perpendicular Tangents

In mathematics, when we talk about mutually perpendicular tangents, we're referring to two tangents that meet the circle at right angles, intersecting at an external point. This concept has a broader implication as it helps define a new circle encompassing said path of external points.
From our problem, the provided circle \(x^2 + y^2 = 9\) with center \((0,0)\) sees this intersection locus defined as another circle with larger radius - a shift intrinsic to all mutually perpendicular tangents. Visualizing this can be insightful:

• Perpendicular tangents define a diameter of sorts, with intersections forming a locus twice the radius of the original circle.
• This geometric approach facilitates defining new circle equations - linking mutual perpendicularity to coaxial circle systems.

Understanding these intersections provides grounding in forming intricate circle systems and their loci, involving tangents working in harmony across geometric spaces.