the standard equation of an ellipse & the general equation of a circle are respectively given by the equation

E : X² /A² +Y²/B^{2 = 1}

^and

^{C = x2+y2+2gx+2fy+c}

^{then the equation E+K C =0,K≠0 (k is some constant)}

^{i.e, (X2/A2+y2/B2-1)+K(x2+y2+2gx+2fy+c)=0}

^{represents a curve which passes through the common points of the ellipse &the circile}

^{Ques: if common chords of E & C are inclined at angles φ & Ψ with the axis of E,then}

^{(a) φ+ Ψ=pie(180)}

^{(b) φ+ Ψ = 2tan-1(B/A)}

^{(c) φ+ Ψ = 900}

( d) none

To tackle the problem regarding the relationship between the common chords of an ellipse and a circle, we need to delve into some geometry and algebra. The equations you've provided for the ellipse and the circle are foundational in understanding their intersections and the properties of the chords formed by these intersections.

Understanding the Equations

The standard equation of an ellipse is given by:

E: X²/A² + Y²/B² = 1

Here, A and B are the semi-major and semi-minor axes, respectively. This equation describes an ellipse centered at the origin.

The general equation of a circle is:

C: x² + y² + 2gx + 2fy + c = 0

In this equation, the terms involving g and f help define the center of the circle, while c relates to its radius.

Combining the Equations

When we combine these two equations as:

E + K*C = 0, where K is a non-zero constant, we are essentially creating a new equation that represents a curve passing through the common points of the ellipse and the circle. This curve will have properties influenced by both the ellipse and the circle.

Common Chords and Their Angles

The common chords of the ellipse and circle are the lines that intersect both shapes at two points. The angles φ and ψ that these chords make with the axis of the ellipse are crucial for understanding their geometric relationship.

Analyzing the Angles

To find the relationship between φ and ψ, we can use the properties of the ellipse and circle. The inclination of the common chords can be derived from the geometry of the intersection points. The key insight is that the sum of the angles φ and ψ relates to the axes of the ellipse and the geometry of the shapes involved.

• (a) φ + ψ = π (180 degrees)
• (b) φ + ψ = 2tan⁻¹(B/A)
• (c) φ + ψ = 90°
• (d) none

Determining the Correct Option

From the properties of conics, particularly the ellipse and circle, we can derive that the angles between the common chords are related to the axes of the ellipse. Specifically, the correct relationship is:

φ + ψ = 2tan⁻¹(B/A)

This result arises from the geometric properties of the ellipse and the nature of the angles formed by the intersection of the two curves.

Final Thoughts

In summary, the relationship between the angles of the common chords of the ellipse and the circle is not only a fascinating aspect of geometry but also highlights the interconnectedness of different conic sections. The correct answer to your question is option (b): φ + ψ = 2tan⁻¹(B/A).