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Assertion (A): The image of x^2 25 + y^2 16 =1 in the line x+y=10 is (x-10)^2 16 + (y-10)^2 25 =1 . Reason (R): The image of a curve ' C ' in a line L is the locus of the image of every point of C with respect to the line L . The correct option among the following is:

Options

  1. A(A) is true, (R) is true and (R) is the correct explanation for (A)
  2. B(A) is true, (R) is true but (R) is not the correct explanation for (A)
  3. C(A) is true but (R) is false
  4. D(A) is false but (R) is true

Correct answer

A. (A) is true, (R) is true and (R) is the correct explanation for (A)

Step-by-step solution

Given equation of ellipse is x^2 25 + y^2 16 =1 The image of the ellipse in the line x+y-10 will be (x-h)^2 16 + (y-k)^2 25 =1 Centre of original equation is (0,0) and other equation is (h, k) Midpoint is ( h 2 , k 2 ) Then, x+y-10=0 aligned & h 2 + k 2 =10 & h+k=20 aligned Now, Given equation is perpendicular to the line joining the coordinates (0,0) &(h, k) So, k h (-1)=(-1) From (i), h+k=20 aligned & 2 h=20 & h=10 aligned Then, k=10 Equation of ellipse = (x-10)^2 16 + (y-10)^2 25 =1 It is obtained by the image o

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