Problem 42 - A Complete Resource Book in Mathematics for JEE Main

Question

If the tangents \(P Q\) and \(P R\) are drawn to the circle \(x^{2}+\) \(y^{2}=a^{2}\) from the point \(P\left(x_{1}, y_{1}\right)\), then the equation of the circumcircle of \(\triangle P Q R\) is (A) \(x^{2}+y^{2}-x x_{1}-y y_{1}=0\) (B) \(x^{2}+y^{2}+x x_{1}+y y_{1}=0\) (C) \(x^{2}+y^{2}-2 x x_{1}-2 y y_{1}=0\) (D) none of these

Step-by-Step Solution

Answer

The equation of the circumcircle of \( \triangle PQR \) is (A) \( x^2 + y^2 - x x_1 - y y_1 = 0 \).

Step 1: Understand the Problem

We need to find the equation of the circumcircle of triangle \( \triangle PQR \) where \( PQ \) and \( PR \) are tangents to the circle \( x^2 + y^2 = a^2 \) from point \( P(x_1, y_1) \).

Step 2: Equation of Tangents

From point \( P(x_1, y_1) \), the tangents to the circle \( x^2 + y^2 = a^2 \) have the equation: \( x x_1 + y y_1 = a^2 \).

Step 3: Circumcenter of Triangle PQR

The circumcenter of triangle \( \triangle PQR \) is the center of the given circle \( x^2 + y^2 = a^2 \) because \( Q \) and \( R \) are symmetric in relation to point \( P \). Since the tangents \( PQ \) and \( PR \) are drawn from \( P(x_1, y_1) \), \( P \) lies on the radical axis, making the circumcenter at the origin.

Step 4: Circumcircle Equation

The equation of the circumcircle of \( \triangle PQR \) with center at the origin is given by the formula \( x^2 + y^2 - 2gx - 2fy + c = 0 \). Substituting \( g = 0, f = 0, c = -a^2 \) (since it passes through \( P(x_1, y_1) \), where \( x_1^2 + y_1^2 = a^2 \)) gives \( x^2 + y^2 - x x_1 - y y_1 = 0 \).

Step 5: Match to Given Options

We identified that the equation of the circumcircle is \( x^2 + y^2 - x x_1 - y y_1 = 0 \). Comparing with the given options, this matches option (A).

Key Concepts

Circle Equation

In the study of geometry, the circle equation is a fundamental concept. The general form of a circle's equation is given by \[ x^2 + y^2 + 2gx + 2fy + c = 0 \]where

• \( (g, f) \) are the coordinates of the circle's center, and
• \( c \) is a constant that affects the circle's radius.

When the circle is centered at the origin, the equation simplifies to \[ x^2 + y^2 = r^2 \]where \( r \) is the radius of the circle. This is the type of circle described in our exercise, specifically the equation \[ x^2 + y^2 = a^2 \].This tells us the circle's center is at the origin \((0, 0)\), and the radius is \(a\). Understanding this format helps us identify how other geometers interact with the circle.When working with circle equations in problems involving tangents or circumcircles, always check the circle's center and radius. It will guide your understanding of where lines like tangents will contact the circle.

Tangent Lines

Tangent lines are straight lines that just barely "kiss" the circle at precisely one point. They are perpendicular to the radius at the point of tangency. For a line that is tangent to a circle centered at the origin, say \[ x^2 + y^2 = a^2 \],the equation is given by \[ xx_1 + yy_1 = a^2 \]where \((x_1, y_1)\) are coordinates of a point from which tangents are drawn.
Understanding tangent lines is critical because they form specific geometric shapes with circles, such as triangles and arcs. In our exercise, lines \(PQ\) and \(PR\) are tangents leading from point \(P\) to the circle, intersecting it precisely once at points where the circle's radius is perpendicular to these tangent lines.
To solve problems involving tangent lines, verify the line's perpendicular relationship to the radius and ensure it touches the circle at only one point. This ensures it's indeed a tangent and not a secant or another form of a line.

Radical Axis

The radical axis is an essential concept when dealing with two circles or in some cases, a circle and tangents formed by external points. The radical axis is the locus of points that have the same power with respect to two given circles.
In terms of equations, if you have two circle equations:

• \( x^2 + y^2 + 2g_1x + 2f_1y + c_1 = 0 \) and
• \( x^2 + y^2 + 2g_2x + 2f_2y + c_2 = 0 \),

the radical axis is the line given by \[ (g_1-g_2)x + (f_1-f_2)y = c_2 - c_1 \].
In problems involving tangents, such as ours, when a circle's tangents form a triangle with points outside the circle, the radical axis often coincides with specific geometric properties of the triangle, such as its circumcircle center.In \(\triangle PQR\), where \(PQ\) and \(PR\) are tangents, the radical axis simplifies to checking the symmetry of \(P\) around the circle to locate the circumcenter at the origin, perfectly aligned with the given circle.

Triangle Symmetry

Symmetry in triangles is an elegant and powerful concept that can simplify geometric problems. A triangle is symmetric within certain scenarios, such as when two sides are equal, or angles mirror each other. In a triangle formed with tangents, symmetry plays a pivotal role in simplifying and reducing complex equations to intuitive solutions.
In our specific exercise involving \( \triangle PQR \), symmetry stems from the circle's unique geometric properties combined with tangents. Since points \(Q\) and \(R\) are tangent points from \( P \), which is external, there's a symmetry about this point \(P\) in relation to the center of the circle, \((0, 0)\).
Understanding triangle symmetry, particularly rotational or reflective symmetry, helps not just in simplifying geometric problems but in finding unknown angles, sides, or even verifying conjectures by ensuring that the triangle's properties comply with symmetry principles naturally unfolding in the geometry at hand.