Statement I Straight line `x+y=k` touch the parabola `y=x-x^2`, if k=1.
Statement II Discriminant of `(x-1)^2=x-x^2` is zero.

Options

A. Statement I is true, Statement II is true , Statement II is a correct explanation for statement I.
B. Statement I is true, Statement II is true, Statement II is not a correct explanation for Statement I.
C. Statement I is true, Statement II is false.
D. Statement I is false,Statement II is true.

Answer

To solve the problem, we will analyze both statements step by step.

**Step 1: Analyze Statement I**
We have the line given by the equation:
\[ x + y = k \]
For \( k = 1 \), the equation becomes:
\[ x + y = 1 \]
This can be rewritten as:
\[ y = 1 - x \]

We also have the parabola given by:
\[ y = x - x^2 \]

To check if the line touches the parabola, we need to set the two equations for \( y \) equal to each other:
\[ 1 - x = x - x^2 \]

**Step 2: Rearranging the Equation**
Rearranging the equation gives:
\[ x^2 - 2x + 1 = 0 \]
This simplifies to:
\[ (x - 1)^2 = 0 \]

**Step 3: Finding the Discriminant**
The equation \( (x - 1)^2 = 0 \) is a perfect square, meaning it has exactly one solution. The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( D = b^2 - 4ac \). Here, we can identify:
- \( a = 1 \)
- \( b = -2 \)
- \( c = 1 \)

Calculating the discriminant:
\[ D = (-2)^2 - 4 \cdot 1 \cdot 1 = 4 - 4 = 0 \]

Since the discriminant is zero, this confirms that the line touches the parabola.

**Conclusion for Statement I:**
Thus, Statement I is true.

**Step 4: Analyze Statement II**
Statement II claims that the discriminant of the equation:
\[ (x - 1)^2 = x - x^2 \]
is zero.

Rearranging this gives us:
\[ x^2 - 3x + 1 = 0 \]

**Step 5: Finding the Discriminant for Statement II**
Now, we need to find the discriminant of this quadratic equation:
- \( a = 1 \)
- \( b = -3 \)
- \( c = 1 \)

Calculating the discriminant:
\[ D = (-3)^2 - 4 \cdot 1 \cdot 1 = 9 - 4 = 5 \]

Since the discriminant is 5, which is greater than zero, this means that the equation has two distinct real roots.

**Conclusion for Statement II:**
Thus, Statement II is false.

**Final Conclusion:**
- Statement I is true.
- Statement II is false.

**Answer: Statement I is true, Statement II is false.**

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