Question

The curves \( y = x^2 - 1 \) and \( y = 8x - x^2 - 9 \)

• A. intersect at right angles at (2, 3)
• B. touch each other at (2, 3)
• C. intersect at 45\(^\circ\)
• D. intersect at 60\(^\circ\)

Hint:

Touching Curves}
Check point of intersection by solving equations
Verify if the first derivatives (slopes) at the point match
If derivatives match and point lies on both curves, the curves touch

Answer 1

The Correct Option is B

Find the point of intersection: \[ x^2 - 1 = 8x - x^2 - 9 \Rightarrow 2x^2 - 8x + 8 = 0 \Rightarrow (x - 2)^2 = 0 \Rightarrow x = 2 \] Then \( y = 2^2 - 1 = 3 \Rightarrow \text{Point is } (2, 3) \) Next, check slopes: - For \( y = x^2 - 1 \): \( y' = 2x \Rightarrow 2(2) = 4 \) - For \( y = 8x - x^2 - 9 \): \( y' = 8 - 2x \Rightarrow 8 - 4 = 4 \) Slopes are equal \( \Rightarrow \) tangents are same \( \Rightarrow \) curves touch at (2, 3).