Multiple choice

Tangents are drawn to the circles $x^{2} + y^{2} + 4x + 6y - 19 = 0$ and $x^{2} + y^{2} = 9$ from any point on the line $2x + 3y = 5$.then their lengths are equal.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The length of the tangent from a point (x1, y1) to a circle is sqrt(x1^2 + y1^2 + 2gx1 + 2fy1 + c). For the first circle, L1^2 = x1^2 + y1^2 + 4x1 + 6y1 - 19. For the second, L2^2 = x1^2 + y1^2 - 9. If the point lies on 2x + 3y = 5, then 2x1 + 3y1 = 5. Substituting 4x1 + 6y1 = 2(2x1 + 3y1) = 10 into L1^2: L1^2 = x1^2 + y1^2 + 10 - 19 = x1^2 + y1^2 - 9. Since L1^2 = L2^2, the lengths are equal.

AI explanation

The lengths of the tangents drawn from a point to two given circles will be equal if the point lies on their radical axis, which is found by subtracting the equations of the two circles. Subtracting the second circle x squared plus y squared minus 9 equals 0 from the first circle x squared plus y squared plus 4x plus 6y minus 19 equals 0 gives the equation 4x plus 6y minus 10 equals 0, which simplifies to 2x plus 3y equals 5. Since the given line is exactly the radical axis of the two circles, the tangent lengths drawn from any point on this line to both circles must be equal, making the statement true.

Keep practicing — related questions