Question
(d) The graph of \( y=1-\frac{2}{x^{2}} \) cuts the positive \( x \)-axis at \( A \). \( B \) is the point \( (0,-2) \) (i) Write down the co-ordinates of \( A \).
Ask by Mitchell Vargas. in China
Jan 22,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The coordinates of point \( A \) are \( \left( \sqrt{2},\; 0 \right) \).
Solution
To find the coordinates of point \( A \) where the graph of \( y = 1 - \frac{2}{x^2} \) intersects the positive \( x \)-axis, follow these steps:
1. **Intersection with the \( x \)-axis:**
At the \( x \)-axis, the \( y \)-coordinate is 0. So, set \( y = 0 \):
\[
0 = 1 - \frac{2}{x^2}
\]
2. **Solve for \( x \):**
\[
\frac{2}{x^2} = 1 \implies x^2 = 2 \implies x = \sqrt{2} \quad (\text{since } x > 0)
\]
3. **Coordinates of \( A \):**
\[
A = \left( \sqrt{2}, \, 0 \right)
\]
**Answer:**
(i) The coordinates of \( A \) are \( \left( \sqrt{2},\; 0 \right) \).
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Simplify this solution The Deep Dive
To find the coordinates of point \( A \) where the graph cuts the positive \( x \)-axis, we need to set \( y = 0 \) in the equation \( y = 1 - \frac{2}{x^2} \). Setting \( y \) to zero: \[ 0 = 1 - \frac{2}{x^2} \] Rearranging gives: \[ \frac{2}{x^2} = 1 \] This simplifies to: \[ x^2 = 2 \] Taking the positive square root (since we are looking for the point on the positive \( x \)-axis): \[ x = \sqrt{2} \] Therefore, the coordinates of point \( A \) are \( A (\sqrt{2}, 0) \).
