Skip to content
Exercise 6.5 · Q24

Q.θ\theta is the acute angle between the lines x2−xy−6y2=0x^2-xy-6y^2=0. Then 2cos⁡θ+3sin⁡θ4sin⁡θ+5cos⁡θ\dfrac{2\cos\theta+3\sin\theta}{4\sin\theta+5\cos\theta} is

(1) 11
(2) −19-\dfrac19
(3) 59\dfrac59
(4) 19\dfrac19
Puducherry TnboardTextbookSubjectiveImportance★★★★★
71% · 92/129 Questions
🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Factor x2−xy−6y2=(x−3y)(x+2y)x^2-xy-6y^2=(x-3y)(x+2y) to get slopes 13,−12\tfrac13,-\tfrac12; the acute angle between them has tan⁡θ=1\tan\theta=1, so θ=45°\theta=45° and the ratio simplifies to 59\dfrac59.

Step 1. Factor the homogeneous pair. x2−xy−6y2=(x−3y)(x+2y)x^2-xy-6y^2=(x-3y)(x+2y) — check: (x−3y)(x+2y)=x2+2xy−3xy−6y2=x2−xy−6y2(x-3y)(x+2y)=x^2+2xy-3xy-6y^2=x^2-xy-6y^2. ✓ So the two lines are

x−3y=0 (slope m1=13)andx+2y=0 (slope m2=−12)x-3y=0 \ \left(\text{slope } m_1=\tfrac13\right) \qquad\text{and}\qquad x+2y=0 \ \left(\text{slope } m_2=-\tfrac12\right)

Step 2. Find the angle between the lines. The angle θ\theta between two lines of slopes m1,m2m_1,m_2 satisfies tan⁡θ=∣m1−m21+m1m2∣\tan\theta=\left|\dfrac{m_1-m_2}{1+m_1m_2}\right|:

tan⁡θ=∣13−(−12)1+13(−12)∣=∣16+13⋅3/31−16∣  ...simplify directly:\tan\theta=\left|\dfrac{\tfrac13-\left(-\tfrac12\right)}{1+\tfrac13\left(-\tfrac12\right)}\right|=\left|\dfrac{\tfrac16+\tfrac13\cdot 3/3}{1-\tfrac16}\right|\; \text{...simplify directly:}

13−(−12)=13+12=2+36=56,1+13(−12)=1−16=56\dfrac13-\left(-\dfrac12\right)=\dfrac13+\dfrac12=\dfrac{2+3}{6}=\dfrac56, \qquad 1+\dfrac13\left(-\dfrac12\right)=1-\dfrac16=\dfrac56

tan⁡θ=∣5/65/6∣=1\tan\theta=\left|\dfrac{5/6}{5/6}\right|=1

(This also follows directly from tan⁡θ=∣2h2−aba+b∣\tan\theta=\left|\dfrac{2\sqrt{h^2-ab}}{a+b}\right| with a=1, 2h=−1⇒h=−12, b=−6a=1,\ 2h=-1\Rightarrow h=-\tfrac12,\ b=-6: h2−ab=14+6=254h^2-ab=\tfrac14+6=\tfrac{25}{4}, tan⁡θ=∣2⋅521−6∣=∣5−5∣=1\tan\theta=\left|\dfrac{2\cdot\tfrac52}{1-6}\right|=\left|\dfrac{5}{-5}\right|=1.)

Step 3. Determine θ\theta. Since θ\theta is the acute angle and tan⁡θ=1\tan\theta=1, we get θ=45°\theta=45°, so sin⁡θ=cos⁡θ=22\sin\theta=\cos\theta=\dfrac{\sqrt2}{2}. …

Start your 14-day free trial to unlock the full solution →

Unlock everything free for 14 days

  • Full step-by-step solutions
  • Concept-first explanations
  • Methods, shortcuts & mistakes
  • PYQ mapping + timed mock tests
Start 14-day free trial →See Plans & Pricing →

Full access for 14 days. No credit card required.

←Previous question · 6.5.4Q23 — If one of the lines given by 6x²-xy+4cy²=0 is 3x+4y=0, then c equals (…→Next question · 6.5.4Q25 — The equation of one of the lines represented by the equation x²+2xycot…