Problem 73 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(S(x, y)=0\) represents a circle. The equation \(S(x, 2)=\) 0 gives two identical solutions \(x=1\) and the equation \(S(1, y)=0\) gives two distinct solutions \(y=0,2\). The equation of the circle is (A) \(x^{2}+y^{2}+2 x+2 y+1=0\) (B) \(x^{2}+y^{2}+2 x+2 y-1=0\) (C) \(x^{2}+y^{2}-2 x-2 y+1=0\) (D) none of these

Step-by-Step Solution

Answer

The equation of the circle is (C) \(x^2 + y^2 - 2x - 2y + 1 = 0\).

Step 1: Analyze the first condition

The equation \(S(x, 2) = 0\) gives the identical solution \(x = 1\). This means that if we substitute \(y = 2\) into the equation of the circle, \((x-1)^2 = 0\), which implies the circle is tangent to the line \(y = 2\), giving multiple roots for a fixed \(y = 2\). Thus, this indicates we have a perfect square on the left side, implying that the center \(x\) coordinate plus some part of the equation equals 1.

Step 2: Analyze the second condition

The equation \(S(1, y) = 0\) provides distinct solutions \(y = 0, 2\). This implies when substituting \(x = 1\) into the equation of the circle, we get a quadratic equation in \(y\) with roots at \(y = 0\) and \(y = 2\). Therefore, the equation can be transformed into \((y)(y-2)=0\).

Step 3: Combine the conditions to form possible circle equation

Given the conditions, the general form of the circle equation is \((x-1)^2 + (y)(y-2) = 0\), which can be expanded to \( (x-1)^2 + y^2 - 2y = 0\). Expanding \((x-1)^2\) gives \((x^2 - 2x + 1)\). Thus, \(x^2 - 2x + 1 + y^2 - 2y = 0\).

Step 4: Simplify and match

Simplify the circle equation \(x^2 - 2x + 1 + y^2 - 2y = 0\) to \(x^2 + y^2 - 2x - 2y + 1 = 0\). Compare it with given options. This matches with the option (C).

Key Concepts

Tangent Line

In the context of a circle, a tangent line is a straight line that touches the circle at exactly one point. This touch point is called the point of tangency. When a circle is tangent to a line at a particular point, it means that the circle just 'kisses' the line there without crossing it. In our circle equation analysis, when we substitute a specific value for either variable (here, \(y = 2\)), and the resulting equation has identical roots (like \(x = 1\)), it indicates that the circle is tangent to the line at that fixed value.

Some key points to note about tangent lines:

• A tangent line is perpendicular to the radius at the point of tangency.
• There is only one tangent line for any given point on a circle.
• They play a crucial role in understanding and deriving equations of circles in coordinate geometry.

Tangency in equations signifies distinct behaviors of variables, often leading to observations such as identical solutions.

Quadratic Equation

A quadratic equation is a second-degree polynomial equation of the form \(ax^2 + bx + c = 0\). In our circle problem, when analyzing \(S(1, y) = 0\), substituting \(x = 1\) gives us a quadratic equation in \(y\), characterized by having two solutions \(y = 0\) and \(y = 2\).

This indicates that when moving along the vertical line where \(x = 1\), the circle intersects the line at two points, corresponding to the two solutions of the quadratic equation. Key characteristics of quadratic equations include:

• They can have two, one, or no real solutions.
• The solutions are determined by the discriminant \(b^2 - 4ac\).
• They graph as parabolas, opening either upwards or downwards based on the sign of coefficient \(a\).

In situations involving circles, solving quadratic equations helps find intersection points or conditions like tangency.

Roots of Equation

The roots of an equation are the values of the variable that satisfy the equation, turning it into a true statement (i.e., they make the equation equal to zero). In our context, solving the modified circle equations \( (x-1)^2 + (y-0)(y-2) = 0 \) and analyzing their solutions helps us understand various properties of the circle.

Roots can be interpreted in different contexts:

• Real and Distinct: Two different solutions, indicating two intersection points.
• Identical or Repeated: Solutions that are the same, indicating tangency or touching at a single point.
• Complex: Non-real solutions, suggesting no intersection in the real plane.

In circle-related problems, the roots often represent coordinates where lines (or specific conditions) meet or touch the circle, offering insights into geometric arrangements.