If the curve ay + x ^{ 2 } = 7 and x ^{ 3 } = y, cut orthogonally at (1, 1), then the value of a is:
(A)1 (B) 0 (C) –6 (D) 6
Last updated at Dec. 4, 2021 by Teachoo
Transcript
Question 10 If the curve ay + x2 = 7 and x3 = y, cut orthogonally at (1, 1), then the value of a is: 1 (B) 0 (C) –6 (D) 6 Two curves cut each other orthogonally means the cut each other at right angles And, if two curves cut at right angles, then their tangents are perpendicular at point of intersection Finding Slope of Tangents of both curves at (1, 1) Slope of tangent of First Curve 𝒂𝒚+𝒙=𝟕 Differentiating w.r.t. x a 𝑑𝑦/𝑑𝑥 + 2𝑥 = 0 𝒅𝒚/𝒅𝒙 = (−𝟐𝒙)/𝒂 At (1, 1), putting x = 1, y = 1 𝑑𝑦/𝑑𝑥=(−2(1))/𝑎 𝑑𝑦/𝑑𝑥=(−2)/𝑎 ∴ Slope =(−𝟐)/𝒂 Slope of tangent of Second Curve 𝒙^𝟑 = 𝒚 Differentiating w.r.t. x 3𝑥^2 = 𝑑𝑦/𝑑𝑥 𝒅𝒚/𝒅𝒙 = 𝟑𝒙^𝟐 At (1, 1), putting x = 1, y = 1 𝑑𝑦/𝑑𝑥= 3(1)^2 𝑑𝑦/𝑑𝑥= 3 ∴ Slope =𝟑 Now, Since tangents are perpendicular Product of the two slopes = −1 (−𝟐)/𝒂 × 3 = −1 −6/𝑎 = −1 −6=−𝑎 𝒂=𝟔 Hence, value of 𝑎 is 6 So, the correct answer is (D)
NCERT Exemplar - MCQs
Question 2
Question 3 Important
Question 4
Question 5
Question 6 Important
Question 7 Important
Question 8
Question 9 Important
Question 10 You are here
Question 11 Important
Question 12 Important
Question 13
Question 14
Question 15
Question 16 Important
Question 17 Important
Question 18 Important
Question 19
Question 20 Important
Question 21 Important
Question 22 Important
Question 23
Question 24 Important
Question 25 Important
Question 26
Question 27
Question 28 Important
Question 29 Important
Question 30 Important
NCERT Exemplar - MCQs
About the Author