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Mathematics · Quadratic Equations · NCERT Examples

Example 8

A complete, independently verified solution for NCERT Example 8.

Mathematics · Chapter 4 · NCERT Examples

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NCERT Example

Example 8

quadratic formula · discriminant · word problem

A pole is to be placed on the boundary of a circular park of diameter 13 m so that its distances from diametrically opposite gates differ by 7 m. Determine whether this is possible and find the distances.

A circular park with diameter AB, point P on the circumference, and right triangle APB.
Figure 4.2: the pole P and diametrically opposite gates A and B.

Solution

Let BP = x m, so AP = x + 7 m. Since angle APB is 90°:

(x+7)2+x2=132(x+7)^2+x^2=13^2
x2+7x−60=0x^2+7x-60=0
a=1,b=7,c=−60a=1,\quad b=7,\quad c=-60
D=72−4(1)(−60)=289D=7^2-4(1)(-60)=289
x=−7±2892=5or−12x=\frac{-7\pm\sqrt{289}}2=5 \text{or} -12

Distance cannot be negative, so −12 is rejected. Thus BP = 5 m and AP = 12 m; 5² + 12² = 13² verifies the geometry.

Final answer

Yes. The pole is 5 m from one gate and 12 m from the other.