Question

.If (x cos theta)/(a)+(y sin theta)/(b)=1 and (ax)/(cos theta)-(by)/(sin theta)=a^(2)-b^(2) prove that (x^(2))/(a^(2))+(y^(2))/(b^(2))=1​

Answer 1

Given: (x cos theta)/(a)+(y sin theta)/(b) = 1 and (ax)/(cos theta)-(by)/(sin theta)=a²-b²

To find: Prove that (x²)/(a²) + (y²)/(b²) = 1​

Solution:

• Now we have given  (x cos theta)/(a)+(y sin theta)/(b) = 1

• Consider:

                x / a cos theta = y / b sin theta = k

                x = ak cos theta ................(i)

                y = bk sin theta................(ii)

• Now we have :

                (ax)/(cos theta)-(by)/(sin theta) = a²-b²

• Consider LHS, we have:

                (ax)/(cos theta)-(by)/(sin theta)

• Putting (i) and (ii), we get:

                (a x ak cos theta)/(cos theta)-(b x bk sin theta)/(sin theta)

                a²k - b²k

                (a² - b²)k

• Now RHS is a² - b².

• Comparing both, we get:

                (a² - b²)k = a² - b²

                k = 1.

• Putting k in (i) and (ii), we get:

                x = ak cos theta = a cos theta

                x/a = cos theta

                y = bk sin theta = b sin theta

                y/b = sin theta

• Now Squaring both terms and adding them, we get:

                (x/a)² + (y/b)² =  sin² theta + cos² theta

                (x/a)² + (y/b)² = 1

• Hence proved.

Answer:

             So we have proved that (x/a)² + (y/b)² = 1

Answer 2

Answer:

Step-by-step explanation: