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)

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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