Learn to Create a Robotic Device Using Arduino in the Free Webinar. Register Now
One of our academic counsellors will contact you within 1 working day.
Click to Chat
1800-1023-196
+91-120-4616500
CART 0
Use Coupon: CART20 and get 20% off on all online Study Material
Welcome User
OR
LOGIN
Complete Your Registration (Step 2 of 2 )
Free webinar on Robotics (Block Chain) Learn to create a Robotic Device Using Arduino
30th Jan @ 5:00PM for Grade 1 to 10
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).
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 = a²Replace S(α, β) by (x,y) to get the locus. => (x² + y²) a² = (x² - y²)²
Post Question
Dear , Preparing for entrance exams? Register yourself for the free demo class from askiitians.
points won -