Problem 85 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The circle \(x^{2}+y^{2}-4 x-4 y+4=0\) is inscribed in a triangle which have two of its sides along the coordinate axes. If the locus of the circumcentre of the triangle is \(x+y-x y+k \sqrt{x^{2}+y^{2}}=0\), then \(k\) is equal to (A) 1 (B) \(-1\) (C) 2 (D) none of these

Step-by-Step Solution

Answer

The value of \(k\) is none of these options (i.e., \(k=0\)).

Step 1: Identify Components of the Circle

First, let's rewrite the equation of the circle in a standard form to identify its center and radius. The given circle equation is \(x^2 + y^2 - 4x - 4y + 4 = 0\). Completing the square for both \(x\) and \(y\) parts: \((x-2)^2 - 4 + (y-2)^2 - 4 + 4 = 0\), simplifying gives \((x-2)^2 + (y-2)^2 = 4\). The circle is centered at \((2, 2)\) with a radius of 2.

Step 2: Determine the Equation of the Triangle

Since the circle is inscribed within the triangle and has sides along the coordinate axes, the perpendicular from the center of the circle \((2, 2)\) to both axes would be equal to the radius, 2. Thus, the triangle's vertices are \((4, 0)\), \((0, 4)\), and \((0, 0)\).

Step 3: Understand the Circumcentre's Properties and Locus

For a right-angled triangle with legs meeting at the origin, the circumcentre is at the midpoint of the hypotenuse. For the given vertices \((4, 0)\) and \((0, 4)\), the hypotenuse is the line segment between these two points, so its midpoint is \((2, 2)\). The locus of such a circumcenter for different cases is given by \(x+y-xy+k \sqrt{x^2+y^2}=0\).

Step 4: Substitute Known Values into the Locus Equation

Substitute \(x = 2\) and \(y = 2\) into the locus equation: \(2 + 2 - 2 \, \cdot \, 2 + k \sqrt{2^2 + 2^2} = 0\), which simplifies to \(4 - 4 + k \times 2\sqrt{2} = 0\). This results in \(2k\sqrt{2} = 0\), leading to \(k = 0\). Since none of the options directly match \(k = 0\), the answer is \(\text{none of these}\).

Key Concepts

Circle Geometry

Circle geometry deals with the properties and figures involving circles. It's a fundamental concept in geometry, where we focus on topics such as the circle's center, radius, diameter, and more. Understanding a circle's equation is pivotal. For instance, a circle given by the equation

• \(x^2 + y^2 - 4x - 4y + 4 = 0\)

can be rewritten in its standard form by completing the square.The standard form

• \((x-h)^2 + (y-k)^2 = r^2\)

reveals the circle's center

• \((h, k)\)

and the radius \(r\). In our problem, this circle is centered at

• \((2, 2)\)

with a radius of 2, implying every point on its circumference is 2 units away from the center. This understanding further helps in positioning the circle within the constraints of other shapes like triangles. Recognizing how circles relate to squares or triangles is key to solving many geometric problems. Each position of the circle can bring different results, which is why circle geometry is essential for more complex coordinate geometry tasks.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, bridges algebra and geometry. It allows us to use algebraic formulas to describe geometrical shapes and their properties on the coordinate plane.In this exercise, the given triangle has its sides along the coordinate axes. Thus, its vertices can be conveniently expressed in terms of coordinates. For our triangle:

• One vertex is at the origin
  • \((0,0)\)
• The other two vertices, based on the circle's properties, are
  • \((4, 0)\) and \((0, 4)\).

With coordinate geometry, we can easily find the lengths of the segments between these points or calculate midpoints. For example, the midpoint of the hypotenuse between

• \((4, 0)\) and \((0, 4)\)

is found by averaging the x and y coordinates, resulting in

• \((2, 2)\).

Using coordinate geometry techniques enables precise calculation of loci, distances, and the position of the circumcentric relative to the triangle. This powerful method allows us to solve intricate problems involving shapes and algebraic equations seamlessly.

Right-Angled Triangles

Right-angled triangles are a type of triangle where one angle is exactly 90 degrees. This property makes it one of the fundamental shapes in geometry owing to its simplicity and the power of the Pythagorean theorem. In the context of this exercise, the triangle has its right-angle at the origin

• \((0,0)\)

and its other vertices along the x and y axes as

• \((4,0)\)
  \((0,4)\).

Within any right triangle, the hypotenuse is the side opposite the right angle and is the longest side. Here, the hypotenuse extends between

• \((4, 0)\)
• and \((0, 4)\).

For right-angled triangles like this, the circumcentre is at the midpoint of the hypotenuse. Thus, the midpoint calculation becomes critical, as it provides the location of the circle's circumcentre. In coordinate geometry, being able to confidently determine these positions enables us to solve problems about how triangle vertices and circumcentres relate to other elements, like the inscribed circle. Right-angled triangles act as a foundation for understanding more complex polygons and the interactions between geometric figures in various contexts.