Let T be a circle with diameter A B 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 A B through P intersecting O M at Q .

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Let T be a circle with diameter A B 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 A B through P intersecting O M at Q . The locus of Q as M varies over T is

Quantrex Academy · Accepted Answer

Correct answer: B. a parabola

Let the equation of circle x 2 + y 2 = r 2 Let M r cos θ , r sin θ Equation of tangents at M r cos θ , r sin θ is r x cos θ + r y sin θ = r 2 ⇒ x cos θ + y sin θ = r Tangent is passes through P . ∴ x cos θ + y sin θ = r ⇒ y = r 1 - cos θ sin θ = r tan θ 2 P Q is parallel to A B . ∴ k = r tan θ 2 O Q is normal of circle. ∴ Slope of normal = k h = tan θ ⇒ k h = 2 tan θ 2 1 - tan 2 θ 2 ⇒ k h = 2 k r 1 - k 2 r 2 ∵ tan θ 2 = k r ⇒ 2 h r = r 2 - k 2 ⇒ k 2 = r 2 2 h r Locus of Q is y 2 = - 2 r x - r 2 which is parabola.

Let T be a circle with diameter A B 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 A B through P intersecting O M at Q . The locus of Q as M varies over T is

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