Let a line parallel to z-axis passing through a point P(3,4,a) intersects the plane x−2y+2z=a2+4a+1 at Q where a ∈ R. If least area of ΔOPQ is equal to (pq) Where p and q are co-prime numbers, then the value of (p+q).

Q≡ (3,4,λ+a)3−8+2(λ+a)=a2+4a+12λ=a2+2a+6⇒λ=12(a2+2a+6)Ar(ΔOPQ)=12×5×12(a2+2a+6)=54[(a+1)2+5]∴ Least area =254≡pq⇒p+q=29

Book Online Demo

Check Your IQ

Try Test

Courses

Dropper NEET Course Dropper JEE Course Class - 12 NEET Course Class - 12 JEE Course Class - 11 NEET Course Class - 11 JEE Course Class - 10 Foundation NEET Course Class - 10 Foundation JEE Course Class - 10 CBSE Course Class - 9 Foundation NEET Course Class - 9 Foundation JEE Course Class -9 CBSE Course Class - 8 CBSE Course Class - 7 CBSE Course Class - 6 CBSE Course

Offline Centres

Let a line parallel to z-axis passing through a point $P (3, 4, a)$ intersects the plane $x - 2 y + 2 z = a^{2} + 4 a + 1 a t Q w h e r e a \in R .$ If least area of $Δ O P Q$ is equal to $(\frac{p}{q})$ Where p and q are co-prime numbers, then the value of $(p + q)$ .

see full answer

Start JEE / NEET / Foundation preparation at rupees 99/day !!

21% of IItians & 23% of AIIMS delhi doctors are from Sri Chaitanya institute !!

An Intiative by Sri Chaitanya

answer is 29.

(Unlock A.I Detailed Solution for FREE)

Ready to Test Your Skills?

Check your Performance Today with our Free Mock Test used by Toppers!

Take Free Test

Detailed Solution

$Q \equiv (3, 4, λ + a)$

$\begin{array}{l} 3 - 8 + 2 (λ + a) = a^{2} + 4 a + 1 \\ 2 λ = a^{2} + 2 a + 6 \Rightarrow λ = \frac{1}{2} (a^{2} + 2 a + 6) \\ A r (Δ O P Q) = \frac{1}{2} \times 5 \times \frac{1}{2} (a^{2} + 2 a + 6) = \frac{5}{4} [{(a + 1)}^{2} + 5] \\ ∴ L e a s t a r e a = \frac{25}{4} \equiv \frac{p}{q} \\ \Rightarrow p + q = 29 \end{array}$

Watch 3-min video & get full concept clarity

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

How to Join IIT after 10th

🔥 Start Your JEE/NEET Prep at Just ₹1999 / month - Limited Offer! Check Now!

Get Expert Academic Guidance – Connect with a Counselor Today!

Result Producing online coaching for JEE/NEET of south India

Largest All India Test Series for JEE/NEET

Best Books for IIT JEE

Best Books for NEET

How to Join IIT after 10th