✓ Board-verified
Mathematics · 2025 · 5 marks
CBSE 2025 · Region 6 · Set 1 · Q32
(a) There is a circular park of diameter 65 m as shown in the following figure, where AB is a diameter. An entry gate is to be constructed at a point…
There is a circular park of diameter $\displaystyle 65$ m as shown in the following figure, where AB is a diameter.
An entry gate is to be constructed at a point P on the boundary of the park such that distance of P from A is $\displaystyle 35$ m more than the distance of P from B . Find distance of point P from A and B respectively.
Find the smallest value of p for which the quadratic equation $\displaystyle x^{2}-2(\mathrm{p}+1) x+\mathrm{p}^{2}=0$ has real roots. Hence, find the roots of the equation so obtained.
Marking-scheme solution
Let distance of gate at P from point B is \(\displaystyle x\) m
Then distance of gate at P from point A is \(\displaystyle (35+x) \mathrm{m}\)
In right \(\displaystyle \Delta \mathrm{APB}\)
\[\begin{aligned}
& (x+35)^{2}+x^{2}=(65)^{2} \\
& x^{2}+35 x-1500=0 \\
& (x+60)(x-25)=0 \\
& x=25
\end{aligned}
\]
Hence, \(\displaystyle x+35=60\)
Distance of P from A = $\displaystyle 60$ m
Distance of P from B = $\displaystyle 25$ m
For real roots, \(\displaystyle \mathrm{D} \geq 0\)
\[\begin{aligned}
{[-2(\mathrm{p}+1)]^{2}-4 \mathrm{p}^{2} } & \geq 0 \\
\Rightarrow \mathrm{p} & \geq-\frac{1}{2}
\end{aligned}
\]
∴ smallest value of \(\displaystyle \mathrm{p}=-\frac{1}{2}\)
At \(\displaystyle \mathrm{p}=-\frac{1}{2}\) given equation becomes
\[x^{2}-2\left(\frac{-1}{2}+1\right) x+\left(\frac{-1}{2}\right)^{2}=0
\]
\(\displaystyle x^{2}-x+\frac{1}{4}=0\) or \(\displaystyle 4 x^{2}-4 x+1=0\)
\[(2 x-1)(2 x-1)=0
\]
∴ roots are \(\displaystyle \frac{1}{2}, \frac{1}{2}\)
More from Circles
- In the given figure, TA is a tangent to the circle with centre O such that OT=4 cm, ∠ OTA=30^°, then length of TA is:2023
- Assertion: Radius is the smallest distance of a tangent from the centre of the circle. Reason (R): Radius is perpendicular to the tangent.2026
- Assertion: The tangents drawn at the end points of a diameter of a circle, are parallel. Reason (R): Diameter of a circle is the longest chord.2024
- Prove that the parallelogram circumscribing a circle is a rhombus.2024
- A backyard is in the shape of a triangle ABC with right angle at B. AB=7 m and BC=15 m. A circular pit was dug inside it such that it touches the…2024
- PQ and PR are two tangents to a circle with centre O and radius 5 cm. AB is another tangent to the circle at C which lies on OP. If OP=13 cm, then…2026
- In Fig. 1, AB is diameter of a circle centered at O. BC is tangent to the circle at B. If OP bisects the chord AD and ∠ AOP=60^°, then find m ∠ C.…2022
- PQ is tangent to a circle with centre O. If ∠ POR=65^°, then m ∠ PTR is2026
CBSE Class 10 Mathematics past-paper question from the 2025 board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.