Let T be a circle with diameter AB and centre O. Let l be the tangent to T at B. For each point M on T different from A, consider the tangent t at M…

Question

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

MARKS Content Team · Accepted Answer

Correct answer: (B) a parabola. Let the equation of circle x^2+y^2=r^2 Let Mrcosθ,rsinθ Equation of tangents at Mrcosθ,rsinθ is rxcosθ+rysinθ=r^2 ⇒xcosθ+ysinθ=r Tangent is passes through P. ∴xcosθ+ysinθ=r ⇒y=(r1-cosθ)/(sinθ)=rtan(θ)/2 PQ is parallel to AB. ∴k=rtan(θ)/2 OQ is normal of circle. ∴ Slope of normal =k/h=tanθ ⇒k/h=(2tanθ2)/(1-tan^2θ2) ⇒k/h=2kr/(1-k^2r^2) ∵tan(θ)/2=k/r ⇒2hr=r^2-k^2⇒k^2=r^22hr Locus of Q is y^2=-2rx-r/2 which is parabola.

Solution

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