Name: Formation of Quadratic Equation
Uploaded: 2019-07-19
Duration: 2 min 49 s
Description: We learned how to solve a quadratic equation using different methods. But do you know how to form a quadratic equation with the given roots? Let’s learn in detail here.

Question

A pole has to be erected at a point on the boundary of a circular park of diameter

13

metres in such a way that the differences of its distances from two diametrically opposite fixed gates

A

and

B

on the boundary is

7

metres. Is it possible to do so? If yes, at what distances from the two gates should the pole be erected?

Accepted Answer

Let $P$ be the required location of the pole.

Let the distance of the pole from the gate $B$ be $x m$ , i.e., $B P = x m$ . Now the difference of the distances of the pole from the two gates $= A P - B P$ (or, $B P - A P$ ) $= 7 m$ .

Therefore, $A P = (x + 7) m$ .

Now, $A B = 13 m$ , and since $A B$ is a diameter,

$∠ A P B = 90 °$

$A P^{2} + P B^{2} = A B^{2}$ (By Pythagoras theorem)

i.e., ${(x + 7)}^{2} + x^{2} = 13^{2}$

i.e., $x^{2} + 14 x + 49 + x^{2} = 169$

$2 x^{2} + 14 x - 120 = 0$

So, the distance $' x'$ of the pole from gate $B$ satisfies the equation

$x^{2} + 7 x - 60 = 0$

So, it would be possible to place the pole if this equation has real roots. To see if this is so or not, let us consider its discriminant. The discriminant is

$b^{2} - 4 a c = 7^{2} - 4 \times 1 \times (- 60) = 289 > 0$

So, the given quadratic equation has two real roots, and it is possible to erect the pole on the boundary of the park.

Solving the quadratic equation $x^{2} + 7 x - 60 = 0$ , by the quadratic formula, we get

$x = \frac{- 7 \pm \sqrt{289}}{2} = \frac{- 7 \pm 17}{2}$

Therefore, $x = 5$ or $- 12$ .

Since $x$ is the distance between the pole and the gate $B$ , it must be positive. Therefore, $x = - 12$ will have to be ignored. So, $x = 5$ .

Thus, the pole has to be erected on the boundary of the park at a distance of $5 m$ from the gate $B$ and $12 m$ from the gate $A$ .

Important Questions on Quadratic Equations

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