From a point on a 2 x 2 − b 2 y 2 = 1 , tangents are drawn to a 2 x 2 + b 2 y 2 = 1 . Their chord of contact meets a 2 x 2 − b 2 y 2 = 0 at Q , R . Which are true?

For the external point ( a sec θ , b tan θ ) , the chord of contact is x sec θ / a + y tan θ / b = 1 . Solving with y = ± b x / a gives midpoint ( a sec θ , − b tan θ ) , proving B and C. The foot locus is the auxiliary circle, proving D.

Mathematics - Hyperbola Question with Solution | TestHub

MathematicsHyperbolaPole and PolarMedium2 minQB

MathematicsMediummultiple choice

From a point on $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 1$ , tangents are drawn to $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ . Their chord of contact meets $\frac{x ^{2}}{a ^{2}} - \frac{y ^{2}}{b ^{2}} = 0$ at $Q, R$ . Which are true?

Options:(select one or more)

Answer:

B, C, D

Solution:

For the external point $(a sec θ, b tan θ)$ , the chord of contact is $x sec θ / a + y tan θ / b = 1$ . Solving with $y = \pm b x / a$ gives midpoint $(a sec θ, - b tan θ)$ , proving B and C. The foot locus is the auxiliary circle, proving D.

Stream:JEESubject:MathematicsTopic:HyperbolaSubtopic:Pole and Polar

⏱ 2mℹ️ Source: QB

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