TG EAMCET (EAPCET) Mathematic Multiple Choice Questions (MCQs)

TG EAMCET Mathematics - Coordinate Geometry

1. 181. What is the definition of a locus?
   a) The set of all points satisfying a given condition
   b) The straight line joining two points
   c) The distance between two points
   d) The midpoint of two points
   Answer: a) The set of all points satisfying a given condition
2. 182. Which of the following is the equation of a circle with center at the origin and radius r?
   a) \( x^2 + y^2 = r^2 \)
   b) \( x^2 - y^2 = r^2 \)
   c) \( (x - r)^2 + y^2 = 0 \)
   d) \( x^2 + (y - r)^2 = 0 \)
   Answer: a) \( x^2 + y^2 = r^2 \)
3. 183. What is the transformed equation of the axes when the coordinate axes are rotated by \( \theta \)?
   a) \( x' = x \cos \theta - y \sin \theta \), \( y' = x \sin \theta + y \cos \theta \)
   b) \( x' = x \sin \theta + y \cos \theta \), \( y' = x \cos \theta - y \sin \theta \)
   c) \( x' = -x \cos \theta + y \sin \theta \), \( y' = -x \sin \theta - y \cos \theta \)
   d) \( x' = x \cos \theta + y \sin \theta \), \( y' = -x \sin \theta + y \cos \theta \)
   Answer: a) \( x' = x \cos \theta - y \sin \theta \), \( y' = x \sin \theta + y \cos \theta \)
4. 184. Which of the following is the equation of a straight line in the normal form?
   a) \( r = x \cos \theta + y \sin \theta \)
   b) \( x \cos \theta + y \sin \theta = r \)
   c) \( ax + by + c = 0 \)
   d) \( y = mx + c \)
   Answer: b) \( x \cos \theta + y \sin \theta = r \)
5. 185. In the symmetric form of the equation of a straight line, the equation is given by:
   a) \( \frac{x}{a} + \frac{y}{b} = 1 \)
   b) \( y = mx + c \)
   c) \( ax + by + c = 0 \)
   d) \( x^2 + y^2 = r^2 \)
   Answer: a) \( \frac{x}{a} + \frac{y}{b} = 1 \)
6. 186. What is the condition for two straight lines to be concurrent?
   a) The lines must be parallel
   b) The lines must intersect at one point
   c) The lines must have the same slope
   d) The lines must have different slopes
   Answer: b) The lines must intersect at one point
7. 187. The angle between two straight lines with slopes \( m_1 \) and \( m_2 \) is given by the formula:
   a) \( \theta = \tan^{-1} \left( \frac{m_1 + m_2}{1 - m_1 m_2} \right) \)
   b) \( \theta = \frac{m_1 - m_2}{1 + m_1 m_2} \)
   c) \( \theta = \tan^{-1} \left( \frac{m_1 - m_2}{1 + m_1 m_2} \right) \)
   d) \( \theta = \frac{m_1 m_2}{1 + m_1 + m_2} \)
   Answer: a) \( \theta = \tan^{-1} \left( \frac{m_1 - m_2}{1 + m_1 m_2} \right) \)
8. 188. The length of the perpendicular from a point \( P(x_1, y_1) \) to a line \( ax + by + c = 0 \) is given by:
   a) \( \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \)
   b) \( \frac{ax_1 + by_1 + c}{\sqrt{a^2 + b^2}} \)
   c) \( \frac{|ax_1 + by_1|}{\sqrt{a^2 + b^2}} \)
   d) \( \frac{ax_1 + by_1}{\sqrt{a^2 + b^2}} \)
   Answer: a) \( \frac{|ax_1 + by_1 + c|}{\sqrt{a^2 + b^2}} \)
9. 189. What is the distance between two parallel lines \( ax + by + c_1 = 0 \) and \( ax + by + c_2 = 0 \)?
   a) \( \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \)
   b) \( \frac{|c_1 + c_2|}{\sqrt{a^2 + b^2}} \)
   c) \( \frac{|c_1 - c_2|}{a} \)
   d) \( \frac{|c_1 + c_2|}{a} \)
   Answer: a) \( \frac{|c_1 - c_2|}{\sqrt{a^2 + b^2}} \)
10. 190. For a family of straight lines, the condition for the lines to be concurrent is:
    a) The determinant of the coefficient matrix should be non-zero
    b) The determinant of the coefficient matrix should be zero
    c) The slopes of the lines should be equal
    d) The lines must be perpendicular
    Answer: b) The determinant of the coefficient matrix should be zero

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