EllipseHard
Question
PQ is a double ordinate of the ellipse x2 + 9y2 = 9, the normal at P meets the diameter through Q at R, then the locus of the mid point of PR is -
Options
A.a circle
B.a parabola
C.an ellipse
D.a hyperbola
Solution
Equation of normal at P (3cos θ, sin θ) is
3x secθ - ycosecθ = 8 ..... (i)
Now equation of diameter through Q is
3y cosθ + x sinθ = 0 ..... (ii)
Solving (i) & (ii) we get intersection point R,
Let (h, k) be mid point of PR then
2h =
cosθ, 2k = sinθ.
Now cos2θ+ sin2θ = 1
∴
∴ Locus is ellipse.
3x secθ - ycosecθ = 8 ..... (i)
Now equation of diameter through Q is
3y cosθ + x sinθ = 0 ..... (ii)
Solving (i) & (ii) we get intersection point R,
Let (h, k) be mid point of PR then
2h =
cosθ, 2k = sinθ.Now cos2θ+ sin2θ = 1
∴
∴ Locus is ellipse.
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