A. By definition, it is a circle.B. Equation of tangent at (x, y) is γ – y =
(γ - x)
γ – intercept = x - y
, γ - intercept = y - x
given
(x−y.) (y−x.) = C
→ 2xy -
-
x2 = C
→
x2()2 - 2xy
+
y2 + C
= 0
differentiate this with respect to x to get
(x2−xy+C) = 0
→
= 0 or
-
=
on solving y = sin x + C or xy – k
x2 = C a straight line or a hyperbola
C. Given |z – 2 – i| = |z|. |sin arg z|. Let z = x + iy and arg z = 0, then tan θ = y/x
∴
√(x−2)2+(y−1)2 =
√x2+y2 |sin θ|
Squaring and simplifying we get
x2 – 4x – 2y + 5 = 0, a parabola.
D. Let the point be P(
x1,y1) then
−−1 = 0
touches
x2+y2 =
a2c2 =
a2+b2 . The locus is
+ =
, an ellipse