How will the parabola \(y=x^2-2(k-1)x+(k-1)^2\) meet the \(x\)-axis?
Answer and explanation
Correct answer: It will only touch
The equation \(y=x^2-2(k-1)x+(k-1)^2\) can be rewritten as \(y=(x-(k-1))^2\). Hence its roots are equal and the discriminant is \(D=0\). The vertex is \((k-1,0)\), so for every real value of \(k\), the parabola touches the \(x\)-axis at exactly one point. Option D is incorrect because this tangency is not restricted to \(k=1\). Exam tip: When \(D=0\), a quadratic has equal roots and its parabola touches the \(x\)-axis.
Frequently asked questions
What is the correct answer to this question?
It will only touch
Why is this the correct answer?
The equation \(y=x^2-2(k-1)x+(k-1)^2\) can be rewritten as \(y=(x-(k-1))^2\). Hence its roots are equal and the discriminant is \(D=0\). The vertex is \((k-1,0)\), so for every real value of \(k\), the parabola touches the \(x\)-axis at exactly one point. Option D is incorrect because this tangency is not restricted to \(k=1\). Exam tip: When \(D=0\), a quadratic has equal roots and its parabola touches the \(x\)-axis.
Which subject and chapter does this question cover?
This is a Class 10 Mathematics question. Chapter: Quadratic Equations. Topic: Nature of Roots.
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