Find the locus of the pole of the line lx + my + n = 0 with respect to the circle which touches y-axis at the origin.
y(lx – n) = mx2
Let centre of circle be C(h, 0) since circle touches y-axis at the origin.
∴ Radius of the circle = h
∴ Equation of the circle
(x – h)2 + (y – 0)2 = h2
Let the pole be (x1, y1)
∴ Equation of the polar w.r.t circle (1) is
xx1 + yy1 – h (x + x1) = 0
And given line lx + my + n = 0 which is (2):
From first two relations, we get
and from last two relations, we get
∴ Locus of pole is y(lx – n) = mx2
Show that the locus of the point, the powers of which with respect to two given circles are equal, is a straight line.
Find the condition that chord of contact of any external point
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Find the equation of the chord x2 + y2 – 6x + 10y – 9 = 0 which is bisected at (–2, 4).
Find the middle point of the chord intercepted on line lx + my + n = 0 by the circle x2 + y2 = a2.
Find the locus of middle points of chords of the circle x2 + y2 = a2, which subtend right angle at the point (c, 0).
Find the equations of the tangents from the point A(3, 2) to the circle x2+ y2 + 4x + 6y + 8 = 0 .
If two tangents are drawn from a point on the circle x2 + y2 = 50 to the circle x2 + y2 = 25 then find the angle between the tangents.
Find the equation of the diameter of the circle
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Examine if the two circles x2 + y2 – 2x – 4y = 0 and x2 + y2 – 8y – 4 = 0 touch each other externally or internally.