Multiple choice

Chord is drawn to the circle $\displaystyle x^{2}+y^{2}-4x-2y=0$ at a point where it cuts the x-axis whose slope is parallel to the tangent at an origin. The intercept of the chord on y-axis is

  1. $2$
  2. $4$
  3. $6$
  4. $8$
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D Correct answer
Explanation

The circle is (x-2)^2 + (y-1)^2 = 5. It cuts the x-axis at (0,0) and (4,0). The tangent at the origin (0,0) has a slope found by differentiating the circle equation. The chord parallel to this tangent passing through the other point on the x-axis is calculated to find the y-intercept.

AI explanation

For the circle x^2 + y^2 - 4x - 2y = 0, the derivative dy/dx = (4 - 2x) / (2y - 2) gives the tangent slope at the origin (0,0) as -2. The circle cuts the x-axis where y = 0, giving the point (4, 0), and drawing a chord from this point with slope -2 yields the equation y - 0 = -2(x - 4), which simplifies to y = -2x + 8. Setting x = 0 gives the y-axis intercept as 8.

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