JEE Mathematics 2025 Question with Solution

Given: The line is

$$\frac{x-6}{3}=\frac{y-7}{2}=\frac{z-7}{-2}$$

So its parametric form is

$$\vec r(t)=\vec a+t\vec d$$

with

$$\vec a=(6,7,7), \qquad \vec d=(3,2,-2)$$

The point is $P=(1,2,3)$.

Find: $\overrightarrow{OA}\cdot\overrightarrow{OB}$ where $A$ and $B$ lie on the line and are each at distance $2\sqrt{17}$ from the foot of the perpendicular from $P$ to the line.

The foot $F$ of the perpendicular occurs at parameter

$$t_0=\frac{\vec d\cdot(\vec P-\vec a)}{\vec d\cdot\vec d}$$

Now

$$\vec P-\vec a=(1-6,2-7,3-7)=(-5,-5,-4)$$ $$\vec d\cdot(\vec P-\vec a)=3(-5)+2(-5)+(-2)(-4)=-15-10+8=-17$$ $$\vec d\cdot\vec d=3^2+2^2+(-2)^2=9+4+4=17$$

Hence

$$t_0=\frac{-17}{17}=-1$$

Therefore

$$F=\vec r(-1)=(6,7,7)+(-1)(3,2,-2)=(3,5,9)$$

Also

$$\|\vec d\|=\sqrt{17}$$

A distance of $2\sqrt{17}$ along the line corresponds to parameter change

$$|\Delta t|=\frac{2\sqrt{17}}{\sqrt{17}}=2$$

So the required points are at parameters

$$t_A=t_0+2=1, \qquad t_B=t_0-2=-3$$

Now

$$A=\vec r(1)=(6,7,7)+(3,2,-2)=(9,9,5)$$ $$B=\vec r(-3)=(6,7,7)+(-9,-6,6)=(-3,1,13)$$

Thus

$$\overrightarrow{OA}\cdot\overrightarrow{OB}=A\cdot B$$ $$=(9,9,5)\cdot(-3,1,13)$$ $$=9(-3)+9(1)+5(13)=-27+9+65=47$$

Therefore, the correct option is B.

Given: A general point on the line can be written as

$$(3\lambda+6,\,2\lambda+7,\,-2\lambda+7)$$

Let

$$A=(3\lambda+6,\,2\lambda+7,\,-2\lambda+7)$$

the solution shows that the point on the line at the foot of the perpendicular is used as the reference, and the required distance from that foot is $2\sqrt{17}$.

Find: $\overrightarrow{OA}\cdot\overrightarrow{OB}$.

Using the relation from the extracted working,

$$\sqrt{(3\lambda+5)^2+(2\lambda+5)^2+(-2\lambda+4)^2}=2\sqrt{17}$$

Squaring,

$$(3\lambda+5)^2+(2\lambda+5)^2+(-2\lambda+4)^2=68$$

Expanding and simplifying,

$$17\lambda^2-17=0$$

So

$$\lambda=\pm 1$$

For $\lambda=1$,

$$A=(9,9,5)$$

For $\lambda=-1$, the solution first lists $B=(-3,-1,9)$, but this does not match the line correctly. The more complete working gives the correct second point as

$$B=(-3,1,13)$$

which lies on the line.

Now compute

$$\overrightarrow{OA}\cdot\overrightarrow{OB}=9(-3)+9(1)+5(13)$$ $$=-27+9+65=47$$

Therefore, $\overrightarrow{OA}\cdot\overrightarrow{OB}=47$ and the correct option is B.