From the point P(3, 4),chords are drawn to th | vedclass

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(A) Let the midpoint of the chord be $M(h, k)$.The equation of the circle is $x^2 + y^2 - 4x = 0$,which has center $O(2, 0)$.Since $M$ is the midpoint

Vedclass · Accepted Answer

$x^2 + y^2 - 5x - 4y + 6 = 0$

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(A) Let the midpoint of the chord be $M(h, k)$.The equation of the circle is $x^2 + y^2 - 4x = 0$,which has center $O(2, 0)$.Since $M$ is the midpoint of the chord,the line segment $OM$ is perpendicular to the chord $PM$.Therefore,the slope of $OM$ is $m_1 = \frac{k - 0}{h - 2} = \frac{k}{h - 2}$.The slope of $PM$ is $m_2 = \frac{k - 4}{h - 3}$.Since $OM \perp PM$,the product of their slopes is $-1$:$\left(\frac{k}{h - 2}ight) 	imes \left(\frac{k - 4}{h - 3}ight) = -1$$k(k - 4) = -(h - 2)(h - 3)$$k^2 - 4k = -(h^2 - 5h + 6)$$h^2 + k^2 - 5h - 4k + 6 = 0$Replacing $(h, k)$ with $(x, y)$,the locus is $x^2 + y^2 - 5x - 4y + 6 = 0$.

From the point $P(3, 4)$,chords are drawn to the circle $x^2 + y^2 - 4x = 0$. Find the locus of the midpoints of these chords.

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