Problem 10 - A Complete Resource Book in Mathematics for JEE Main

Question

If the lines \(a_{1} x+b_{1} y+c_{1}=0\) and \(a_{2} x+b_{2} y+c_{2}=0\) cut the coordinate axes in concyclic points, then (A) \(a_{1} a_{2}=b_{1} b_{2}\) (B) \(a_{1} b_{1}=a_{2} b_{2}\) (C) \(a_{1} b_{2}=a_{2} b_{1}\) (D) none of these

Step-by-Step Solution

Answer

(C) \(a_{1} b_{2} = a_{2} b_{1}\).

Step 1: Determine points where lines cut the axes

For the line \(a_{1} x + b_{1} y + c_{1} = 0\), find where it cuts the x-axis by setting \(y=0\): \(a_{1} x + c_{1} = 0 \Rightarrow x = -\frac{c_{1}}{a_{1}}\). So, the point is \(\left(-\frac{c_{1}}{a_{1}}, 0\right)\). For the y-axis, set \(x=0\): \(b_{1} y + c_{1} = 0 \Rightarrow y = -\frac{c_{1}}{b_{1}}\). The point is \((0, -\frac{c_{1}}{b_{1}})\). Repeat for the second line.

Step 2: Repeat for the second line

Similarly, for the line \(a_{2} x + b_{2} y + c_{2} = 0\), the points where it cuts the axes are \(\left(-\frac{c_{2}}{a_{2}}, 0\right)\) on the x-axis and \((0, -\frac{c_{2}}{b_{2}})\) on the y-axis.

Step 3: Identify concyclic condition

For the points to be concyclic (on the same circle), the radical axis theorem implies an equivalent condition for the coefficients: The lines joining the intersections with the coordinate axes must satisfy the condition of harmonic division. This involves checking the determinant for equivalent harmonic conjugates in the coefficients: \(a_{1} b_{2} = a_{2} b_{1}\).

Step 4: Conclusion

Since the determinant and harmonic relationships between coefficients needed for the concyclism simplify to \(a_{1}b_{2} = a_{2}b_{1}\), the option (C) is correct. Thus, \(a_{1} b_{2} = a_{2} b_{1}\), which is option (C).

Key Concepts

Equation of a Line

Understanding the equation of a line is fundamental in coordinate geometry. A line in a two-dimensional plane can generally be expressed by the equation \(ax + by + c = 0\). Here, \(a\), \(b\), and \(c\) are constants. This linear equation represents the set of all points \((x, y)\) that lie on the line.

When dealing with such equations, you often need to find points where the line intersects the coordinate axes. For instance:

• To find where a line crosses the x-axis, set \(y = 0\) and solve for \(x\).
• To find where it crosses the y-axis, set \(x = 0\) and solve for \(y\).

Being adept at interpreting and manipulating the equation of a line helps in solving many geometric problems and understanding the placement of points in a Cartesian coordinate system.

Harmonic Division

Harmonic division is a concept in projective geometry where a point divides a line segment into two segments that are in a particular ratio known as a harmonic ratio. Although it sounds complex, it can be understood with simpler ideas from geometry

In the context of the exercise, harmonic division occurs when four points \(A\), \(B\), \(C\), and \(D\) on a line correspond in such a way that the cross-ratio \((A,B;C,D)\) equals \(-1\). This means the points are related through a special symmetry.

Understanding harmonic division involves recognizing such relationships and applying the concept to ensure that any pair of intersecting lines or segments satisfies properties like these. Recognizing harmonic division helps identify key properties of geometric figures, such as concyclic points.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, uses algebra to represent and solve geometric problems. It combines coordinate systems, like the familiar Cartesian plane, with equations to describe geometrical figures and relationships.

Key points in coordinate geometry include:

• The ability to calculate distances and midpoints of segments using coordinates.
• Finding slopes and equations of lines, circles, and other geometrical shapes using algebra.
• Using coordinates to locate specific points such as intersections and points on the axes.

Coordinate geometry allows solving complex geometric problems using algebraic methods. It bridges the gap between algebra and geometry, providing a powerful tool for understanding and applying mathematical concepts. In this exercise, coordinate geometry helps locate where lines intersect the axes, which is essential to finding concyclic points.

Radical Axis Theorem

The Radical Axis Theorem is a fascinating concept in geometry, linking the properties of circles and lines. It states that if you have two circles, the radical axis is the locus of points that have equal power with respect to both circles.

In this exercise, this theorem helps understand when points from two intersecting lines are concyclic, meaning that they all lie on the same circle. Specifically:

• The radical axis serves as the line of intersection of the two circles' power.
• It can be derived from properties involving harmonic division and relationships between line coefficients.

Utilizing the Radical Axis Theorem means identifying when geometrical sets relate through shared properties like equal tangents from points. This understanding reveals deeper connections between lines and circles, aiding in comprehending the conditions required for points to be concyclic.