The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x

Engineering

Mathematics

Chord of Contact Chord with given Mid Point and Pair of Tangents

Question

The locus of the mid-point of the chord of contact of tangents drawn from points lying on the straight line 4x – 5y = 20 to the circle x² + y² = 9 is

20(x² + y²) + 36x – 45y = 0

36(x² + y²) + 20x – 45y = 0

36(x² + y²) – 20x + 45y = 0

20(x² + y²) – 36x + 45y = 0

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Equation of chord of contact with mid‑point of (h, k)

$y - k = \frac{- h}{k} (x - h) \Rightarrow hx + ky = h^{2} + k^{2}$ ........(1)

Equation of chord of contact for point where 4x₁ – 5y₁ = 20 ........(A)

x x₁ + y y₁ = 9 .......(2)

Comparing (1) and (2)

$\frac{x_{1}}{h} = \frac{y_{1}}{k} = \frac{9}{h^{2} + k^{2}}$

$∴ x_{1} = \frac{9 h}{h^{2} + k^{2}}; y_{1} = \frac{9 k}{h^{2} + k^{2}}$

Put equation (A)

$\Rightarrow \frac{36 h}{h^{2} + k^{2}} - 20 = \frac{45 k}{h^{2} + k^{2}}$

$\Rightarrow$ 20 (h² + k²) – 36h + 45k = 0

$∴$ Locus is 20(x² + y²) – 36x + 45y = 0.

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