Question

Find the length of the chord intercepted by the parabola $y={{x}^{2}}+3x$ on the line $x+y=5$.
(a) $3\sqrt{26}$
(b) $2\sqrt{26}$
(c) $\sqrt{26}$
(d) $6\sqrt{2}$

Answer 1

Hint: In this question, first work out the point of intersection of the parabola and the line by substituting either $x$ or $y$ from the equation of line into parabola equation, Now use distance between two point formula to find the length of the chord.

Complete step-by-step answer:

Let us consider the equation of the parabola is $y={{x}^{2}}+3x$ and the equation of the line is $x+y=5$.
To find the point of intersection between them, put $y={{x}^{2}}+3x$ in the equation of the line $x+y=5$.
We have,
$x+{{x}^{2}}+3x=5$
Rearranging the terms, we get
${{x}^{2}}+4x-5=0$
Rearranging the term $4x$ as $5x-x$ , we get
${{x}^{2}}+5x-x-5=0$
Take $x$ common from first two terms and -1 from last two terms, we get
\[x(x+5)-(x+5)=0\]
Now take $(x+5)$ as common term, we get
\[(x+5)(x-1)=0\]
\[(x+5)=0\text{ or }(x-1)=0\]
\[x=-5\text{ or }x=1\]
Which are the required coordinates of the x.
Put x = -5 in the equation of the parabola $y={{x}^{2}}+3x$, we get
$y={{(-5)}^{2}}+3(-5)=25-15=10$
Put x =1 in the equation of the parabola $y={{x}^{2}}+3x$, we get
$y={{(1)}^{2}}+3(1)=1+3=4$
Hence, the points $(-5,10)$ and $(1,4)$ are the points for the intersection of the parabola and the line or chord.
Now to find the length of the chord that means to find the distance between the two points.
The required two points are $({{x}_{1}},{{y}_{1}})=(-5,10)$ and $({{x}_{2}},{{y}_{2}})=(1,4)$.
By using the distance formula, then the length of the chord is given by
The length of the chord = $\sqrt{{{({{x}_{2}}-{{x}_{1}})}^{2}}+{{({{y}_{2}}-{{y}_{1}})}^{2}}}$
The length of the chord = $\sqrt{{{(1+5)}^{2}}+{{(4-10)}^{2}}}$
The length of the chord = \[\sqrt{{{(6)}^{2}}+{{(-6)}^{2}}}\]
The length of the chord = \[\sqrt{36+36}\]
The length of the chord = \[\sqrt{2(36)}\]
The length of the chord = \[6\sqrt{2}\]
Therefore, the correct option for the given question is option (d).

Note: The possibility for the mistake is that you might get confused if the line intersects with the parabola at two different and real points that means the line is the chord of the parabola.