Multiple choice

The locus of midpoints of the chords of contact of $x^2 + y^2 = 2$ from the points on the line $3x + 4y = 10$ is a circle with centre $P$. If $O$ be the origin, then $OP$ is equal to

  1. $2$
  2. $3$
  3. $\dfrac{1}{2}$
  4. $\dfrac{1}{3}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Chord of contact from (x1, y1) to x^2+y^2=2 is xx1+yy1=2. The midpoint (h, k) of this chord satisfies h = 2x1/(x1^2+y1^2) and k = 2y1/(x1^2+y1^2). Since (x1, y1) is on 3x+4y=10, 3(2h/(h^2+k^2)) + 4(2k/(h^2+k^2)) = 10. This simplifies to 6h+8k = 10(h^2+k^2), or h^2+k^2 - 0.6h - 0.8k = 0. The center P is (0.3, 0.4). OP = sqrt(0.3^2 + 0.4^2) = 0.5.

AI explanation

Let a point on the line 3x + 4y = 10 be (h, k), so its chord of contact to the circle x^2 + y^2 = 2 is hx + ky = 2. Let the midpoint of this chord of contact be (X, Y). The equation of a chord with midpoint (X, Y) for the circle x^2 + y^2 = 2 is given by xX + yY = X^2 + Y^2. Since these two equations represent the same line, their coefficients are proportional, so h/X = k/Y = 2 / (X^2 + Y^2). From this, we get h = 2X / (X^2 + Y^2) and k = 2Y / (X^2 + Y^2). Substituting these expressions for h and k back into the line equation 3h + 4k = 10 gives 6X + 8Y = 10(X^2 + Y^2), which simplifies to the locus X^2 + Y^2 - 3X/5 - 4Y/5 = 0. This locus is a circle with centre P at (3/10, 4/10). Therefore, the distance OP from the origin to P is sqrt((3/10)^2 + (4/10)^2) = sqrt(25/100) = 1/2.

Keep practicing — related questions