Question

If 4x2 + py2 = 45 and x2 - 4y2 = 5 cut orthogonally, then the value of p is:

• A. (A) 9
• B. (B) 13
• C. (C) 3
• D. (D) 18

Answer 1

Answer: (A)

(A) 9

Answer 2

Explanation:
Given:Equations of curves, 4x2 + py2 = 45 and x2 - 4y2 = 5 cut orthogonally.We have to find the value of p.Consider,4x2 + py2 = 45Differentiating w.r.t x, we get8x+2 py dydx=0[ Using standard derivatives- 1]⇒dydx=−4xpySimilarly for x2−4y2=5dydx=x4yLet (α,β) be the point of contact.Then, 4α2+pβ2=45...(i)α2−4β2=5 ....(ii)Multiplying (ii) by 4 and subtracting it from (i), we get(p+16)β2=25⇒β2=25p+16...(iii)Using (ii), we getα2=5+4⋅25p+16
=5p+80+100p+16
=5p+180p+16
=5(p+36)p+16Dividing (iv) by (iii), we getα2β2=p+365Now, using tangents and normals, we getm1=(dydx)(α,β)=4αpβand m2=(dydx)(α,β)=α4β: Both curves cut orthogonally, thenm1m2=−1⇒(−4αpβ)⋅α4β=−1⇒1p(p+365)=1⇒5p=p+36⇒p=9Hence, the correct option is (A).