If the curves x^2/a + y^2/b = 1 and x^2/c + y^2/d = 1 intersect at right angles, then which one is the correct relation?

Category: QuestionsIf the curves x^2/a + y^2/b = 1 and x^2/c + y^2/d = 1 intersect at right angles, then which one is the correct relation?
Editor">Editor Staff asked 11 months ago

If the curves x^2/a + y^2/b = 1 and x^2/c + y^2/d = 1 intersect at right angles, then which one is the correct relation?
 
(a) b – a = c – d
 
(b) a + b = c + d
 
(c) a – b = c – d
 
(d) a – b = c + d
 
The question was posed to me by my school principal while I was bunking the class.
 
My doubt stems from Calculus Application in division Application of Calculus of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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1 Answers
Editor">Editor Staff answered 11 months ago

Correct answer is (c) a – b = c – d
 
For explanation: We have, x^2/a + y^2/b = 1  ……….(1) and x^2/c + y^2/d = 1  ……….(2)
 
Let, us assume curves (1) and (2) intersect at (x1, y1). Then
 
x1^2/a + y1^2/b = 1  ……….(3) and x1^2/c + y1^2/d = 1  ……….(4)
 
Differentiating both side of (1) and (2) with respect to x we get,
 
2x/a + 2y/b(dy/dx) = 0
 
Or dy/dx = -xb/ya
 
Let, m1 and m2 be the slopes of the tangents to the curves (1) and (2) respectively at the point (x1, y1); then,
 
m1 = [dy/dx](x1, y1) = -(bx1/ay1) and m2 = [dy/dx](x1, y1) = -(dx1/cy1)
 
By question as the curves (1) and (2) intersects at right angle, so, m1m2 = -1
 
Or -(bx1/ay1)*-(dx1/cy1) = -1
 
Or bdx1^2 = -acy1^2 ……….(5)
 
Now, (3) – (4) gives,
 
bdx1^2(c – a) = acy1^2(d – b)  ……….(6)
 
Dividing (6) by (5) we get,
 
c – a = d – b
 
Or a – b = c – d.