For the parabola \(y=x^2-2kx+(k^2-16)\), at how many distinct points will it intersect the \(x\)-axis?
Answer and explanation
Correct answer: At two distinct points
To find intersections with the \(x\)-axis, set \(y=0\), giving \(x^2-2kx+k^2-16=0\). Its discriminant is \(D=(-2k)^2-4(k^2-16)=64>0\). Therefore, for every real value of \(k\), the equation has two distinct real roots, so the parabola intersects the \(x\)-axis at two distinct points. Exam tip: \(D>0\) indicates two distinct real roots.
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What is the correct answer to this question?
At two distinct points
Why is this the correct answer?
To find intersections with the \(x\)-axis, set \(y=0\), giving \(x^2-2kx+k^2-16=0\). Its discriminant is \(D=(-2k)^2-4(k^2-16)=64>0\). Therefore, for every real value of \(k\), the equation has two distinct real roots, so the parabola intersects the \(x\)-axis at two distinct points. Exam tip: \(D>0\) indicates two distinct real roots.
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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