From points on the circle x 2 + y 2 = a 2 tangents are drawn to hyperbola x 2 -y 2 =a 2 . Prove that locus of middle points of chords of contact is the curve (x 2 -y 2 ) 2 = a 2 (x 2 +y 2 ).

Question

Saurabh Koranglekar · Answer

Arun · Answer

Let a point P(x1, y1) be on the circle C1: x²+ y² = a². --- (1)

Let the tangents to the Hyperbola H1: x² - y² = a² ---- (2), from P be PQ and PR, touching H1 at Q(x2, y2) and R(x3, y3).
So PQ: x *x2 - y * y2 = a², and PR : x * x3 - y * y3 = a² --- (3)

As PQ and PQ pass through P,
x1 * x2 - y1* y2 = a² and x1 * x3 - y1 * y3 = a² --- (4)

Equation of QR - Chord of Contact containing Q & R - is clearly,
x1 * x - y1 * y = a². --- (5)

Midpoint of chord of contact QR is: S(α, β) = [ (x2+x3)/2, (y2 +y3)/2 ].
Adding two equations in (4), we get x1 α - y1 β = a² --- (6)

Equation of chord of contact QR of H1 with its mid point at S(α, β) is given by formula:
T = S1 --- (7)
ie., x α - β y - a² = α² - β² - a²
=> x α - y β = α² - β². --- (8)

Equations (5) and (8) represent the same Chord of contact QR:
So x1 / α = y1 / β = a²/(α² - β²)
or x1 = α a²/(α²+β²) and y1 = β a²/(α² - β²) --- (9)

Substitute (9) in eq (1) to get :
(a⁴ α² + a⁴ β² )/(α² - β²)² = a²
Replace S(α, β) by (x,y) to get the locus.
=> (x² + y²) a² = (x² - y²)²

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From points on the circle x 2 + y 2 = a 2 tangents are drawn to hyperbola x 2 -y 2 =a 2 . Prove that locus of middle points of chords of contact is the curve (x 2 -y 2 ) 2 = a 2 (x 2 +y 2 ).

From points on the circle x² + y² = a² tangents are drawn to hyperbola x²-y²=a². Prove that locus of middle points of chords of contact is the curve (x²-y²)² = a² (x²+y²).

From points on the circle x² + y² = a² tangents are drawn to hyperbola x²-y²=a². Prove that locus of middle points of chords of contact is the curve (x²-y²)² = a² (x²+y²).

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From points on the circle x 2 + y 2 = a 2 tangents are drawn to hyperbola x 2 -y 2 =a 2 . Prove that locus of middle points of chords of contact is the curve (x 2 -y 2 ) 2 = a 2 (x 2 +y 2 ).

From points on the circle x2 + y2 = a2 tangents are drawn to hyperbola x2-y2=a2. Prove that locus of middle points of chords of contact is the curve (x2-y2)2 = a2 (x2+y2).

From points on the circle x2 + y2 = a2 tangents are drawn to hyperbola x2-y2=a2. Prove that locus of middle points of chords of contact is the curve (x2-y2)2 = a2 (x2+y2).

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From points on the circle x² + y² = a² tangents are drawn to hyperbola x²-y²=a². Prove that locus of middle points of chords of contact is the curve (x²-y²)² = a² (x²+y²).

From points on the circle x² + y² = a² tangents are drawn to hyperbola x²-y²=a². Prove that locus of middle points of chords of contact is the curve (x²-y²)² = a² (x²+y²).