Samacheer Kalvi Class 12 Maths Solutions Chapter 5 Two Dimensional Analytical Geometry II Exercise 5.3

Tamilnadu Samacheer Kalvi 12th Maths Solutions Chapter 5 Two Dimensional Analytical Geometry - II Ex 5.3

Question 1. Identify the type of conic section of each of the equations.
(1) \( 2x^2 - y^2 = 7 \)
(2) \( 3x^2 + 3y^2 - 4x + 3y + 10 = 0 \)
(3) \( 3x^2 + 2y^2 = 14 \)
(4) \( x^2 + y^2 + x - y = 0 \)
(5) \( 11x^2 - 25y^2 - 44x + 50y - 256 = 0 \)
(6) \( y^2 + 4x + 3y + 4 = 0 \)
Answer:
(1) For the equation \( 2x^2 - y^2 = 7 \), when compared to the general conic equation \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \), we find \( A = 2 \) and \( C = -1 \). Since \( A \) and \( C \) are not equal and have opposite signs, this indicates a hyperbola. Hyperbolas are commonly seen in the paths of comets and in certain antenna designs.
(2) For the equation \( 3x^2 + 3y^2 - 4x + 3y + 10 = 0 \), we have \( A = 3 \), \( B = 0 \), \( C = 3 \), \( D = -4 \), \( E = 3 \), and \( F = 10 \). Since \( A \) and \( C \) are equal and \( B = 0 \), the conic section is a circle. A circle is a perfectly symmetrical round shape.
(3) For the equation \( 3x^2 + 2y^2 = 14 \), comparing it with the general form gives \( A = 3 \), \( B = 0 \), \( C = 2 \), and \( F = -14 \). Here, \( A \) and \( C \) are different but both have the same positive sign. This condition indicates that the conic section is an ellipse. Ellipses are used in architectural designs and planetary orbits.
(4) For the equation \( x^2 + y^2 + x - y = 0 \), comparing it with the general conic equation, we see that \( A = 1 \) and \( C = 1 \). Also, \( B = 0 \) because there is no \( xy \) term. Since \( A \) equals \( C \) and \( B \) is zero, this equation represents a circle. This is a common form for a circle passing through the origin.
(5) For the equation \( 11x^2 - 25y^2 - 44x + 50y - 256 = 0 \), when comparing it to the general form, we find \( A = 11 \), \( B = 0 \), \( C = -25 \), \( D = -44 \), \( E = 50 \), and \( F = -256 \). Since \( A \) and \( C \) are not equal and have opposite signs, this conic section is a hyperbola. The graph of a hyperbola has two distinct, disconnected branches.
(6) For the equation \( y^2 + 4x + 3y + 4 = 0 \), comparing it with the general conic equation, we observe that there is no \( x^2 \) term, meaning \( A = 0 \). There is also no \( xy \) term, so \( B = 0 \). Since only one of the squared terms is present (\( y^2 \)), and \( B = 0 \), this shape is a parabola. Parabolas are essential in understanding projectile motion and the design of parabolic mirrors.
In simple words: To find the type of conic, look at the numbers in front of the \( x^2 \) (call it \( A \)) and \( y^2 \) (call it \( C \)) terms, and check if there's an \( xy \) term (call it \( B \)). If \( A=C \) and \( B=0 \), it's a circle. If \( A \ne C \) but \( A \) and \( C \) have the same sign (and \( B=0 \)), it's an ellipse. If \( A \) and \( C \) have opposite signs (and \( B=0 \)), it's a hyperbola. If only one of \( A \) or \( C \) is zero (and \( B=0 \)), it's a parabola.

🎯 Exam Tip: To correctly identify conic sections, always compare the given equation with the general form \( Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 \) and analyze the coefficients \( A \), \( B \), and \( C \).