Question
upstudy study bank question image url

If point \( (-1,+-1),(t, t-3) \) and \( (t,-6,3) \) are collinear, pund the value of \( t \)

Ask by Tucker Fowler. in Nigeria
Mar 11,2025

Upstudy AI Solution

Tutor-Verified Answer

Answer

The values of \( t \) that make the points collinear are \( t = -1 \) and \( t = -3 \).

Solution

Sign in to Unlock Answers for Free!

A Learning Platform Trusted by Millions of Real Students and Teachers.

star-icon Unlock

Answered by UpStudy AI and reviewed by a Professional Tutor

Extra Insights

To determine the value of \( t \) for which the points \((-1, -1)\), \((t, t-3)\), and \((t, -6)\) are collinear, we can use the concept of the area of the triangle formed by these points. The area is zero if the points are collinear. Using the determinant formula for the area of the triangle formed by three points \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\): \[ \text{Area} = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \] Substituting in our points: \[ \text{Area} = \frac{1}{2} \left| -1((t-3) - (-6)) + t(-6 - (-1)) + t-3(-1 - (t-3)) \right| = 0 \] Calculating the terms inside the absolute value: 1. For \(-1((t-3) + 6) = -1(t + 3)\). 2. For \(t(-6 + 1) = -5t\). 3. For \(t-3(2 - t) = -t^2 + 5t - 6\). Combining all terms gives: \[ -t - 3 - 5t - t^2 + 5t - 6 = -t^2 - t - 9 = 0 \] This simplifies to: \[ t^2 + t + 9 = 0 \] Calculating the discriminant: \[ D = b^2 - 4ac = 1^2 - 4 \times 1 \times 9 = 1 - 36 = -35 \] Since the discriminant is negative, there are no real solutions for \( t \).

Related Questions

Latest Geometry Questions

Try Premium now!
Try Premium and ask Thoth AI unlimited math questions now!
Maybe later Go Premium
Study can be a real struggle
Why not UpStudy it?
Select your plan below
Premium

You can enjoy

Start now
  • Step-by-step explanations
  • 24/7 expert live tutors
  • Unlimited number of questions
  • No interruptions
  • Full access to Answer and Solution
  • Full Access to PDF Chat, UpStudy Chat, Browsing Chat
Basic

Totally free but limited

  • Limited Solution
Welcome to UpStudy!
Please sign in to continue the Thoth AI Chat journey
Continue with Email
Or continue with
By clicking “Sign in”, you agree to our Terms of Use & Privacy Policy