Multiple choice

A chord of the circle $x^{2} +y^{2}-4x -6y = 0$ passing through the origin subtends an angle $\tan^{-1} \left(\dfrac{7}{4}\right)$ at the point where the circle meets positive y-axis. Equation of the chord is

  1. $ 2x + 3y = 0$
  2. $x + 2y = 0$
  3. $ x -2y = 0 $
  4. $ 2x -3y = 0$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The circle is (x-2)^2 + (y-3)^2 = 13. The center is (2,3). The circle meets the positive y-axis at (0,6). The chord passes through (0,0) and (0,6) is not possible as it is a chord. The line x-2y=0 passes through the origin and satisfies the geometric conditions.

AI explanation

The circle equation x squared plus y squared minus 4x minus 6y equals 0 has its center at (2, 3) and meets the positive y-axis at the point (0, 6). If the chord passing through the origin has a slope m, its equation is y equals mx, and its slope relative to the center is (m minus 3) divided by 2. The line from (0, 6) to the center has a slope of (3 minus 6) divided by (2 minus 0), which is negative 3 divided by 2. Using the tangent formula for the angle between these two lines, the absolute value of (m minus 3) divided by 2 minus the slope of the second line, all over 1 plus their product, yields positive 7 divided by 4. Solving this equation gives m equals positive 2, making the chord equation y equals 2x, or x minus 2y equals 0.

Keep practicing — related questions