Question

The line y = -1 intersects the parabola, \[y={{x}^{2}}-10x+24\] at one point. Find the coordinates of the point of the intersection.
A. (-5, -1)
B. (0, -1)
C. (5, -1)
D. (35, -1)

Answer 1

Hint: To solve this question, what we will do is firstly we will substitute the value of line y = -1 in the equation of the parabola \[y={{x}^{2}}-10x+24\] . Then, we will simplify the quadratic expression obtained and get the value of x – coordinate using quadratic formula.

Complete step-by-step answer:
We have been given the line, y = -1.
Now this line intersects the parabola, \[y={{x}^{2}}-10x+24\] at one point.
Now substitute y = -1 in the equation of parabola.
\[\begin{align}
  & y={{x}^{2}}-10x+24 \\
 & -1={{x}^{2}}-10x+24 \\
\end{align}\]
Rearrange and simplify the above expression.
\[\begin{align}
  & {{x}^{2}}-10x+24+1=0 \\
 & {{x}^{2}}-10x+25=0 \\
\end{align}\]
The above equation is similar to the quadratic equation, whose general form is \[a{{x}^{2}}+bx+c=0\]. Now compare both the equations. We get, a = 1, b = -10, c = 25.
Now substitute these values in the quadratic formula, \[\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\].
\[\dfrac{-\left( -10 \right)\pm \sqrt{{{\left( -10 \right)}^{2}}-4\times 1\times 25}}{2\times 1}=\dfrac{+10\pm \sqrt{100-100}}{2}=\dfrac{10}{2}=5\]
Thus we got the value of x as 5.
Thus x = 5 and y = -1.
We got the point as (5, -1).

So, the line y = -1 intersects parabola, \[y={{x}^{2}}-10x+24\] at (5, -1).
Hence option (c) is the correct answer.

So, the correct answer is “Option (c)”.

Note: Since in this question we need to find the intersection point when one coordinate is already given, we substitute that coordinate to find the remaining one. The equation of line given is a tangent to the given conic since it is touching at only one point. Always remember that general equation of quadratic equation is \[a{{x}^{2}}+bx+c=0\] and value of x can be obtained using quadratic formula which is \[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]. Try to avoid calculation mistakes while solving questions.