KVPY SX SB Exam 2018 Question Paper

Question : 83 of 120

Marks: +1, -0

Let Γ be a circle with diameter AB and centre O. Let l be the tangent to Γ at B. For each point M on Γ different from A, consider the tangent t at M and let it intersect l at P. Draw a line parallel to AB through Pintersecting OM at Q. The locus of Q as M varies over Γ is

an arc of a circle
a parabola
an arc of an ellipse
a branch of a hyperbola

Validate

Solution:

Let eq.

^n

of circle is

x^2 + y^2 = a^ 2

Eq.

^n

of tangent at M is

x \cos θ + y \sin θ = a

Co-ordinate of

P(a,{a(1-\cosθ)}/{\sin})

= P (a, a \tan θ\/2)

Eq.

^{n}

of OM is

y={a\sinθ }/{ a\cosθ}\ x

y = x \tan θ

Let Q(h, k)

k = a \tan θ \/2 : h={a\tanθ\/2}/{\tanθ}

h={a\tanθ\/2}/({{2\tanθ\/2}/{1-\tan^2θ\/2}})

2h = a (1-tan^2 θ \/2)

2h = a (1 -k^2\/a^2)

⇒

2ax = a^2 -y^2

which is parabola

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