Let the point A divide the line segment joining the points P(–1, –1, 2) and Q(5, 5, 10) internally in the ratio r : 1 (r 0). If O is the origin and , then the value of r is : [2025]

Question

Let the point A divide the line segment joining the points P(–1, –1, 2) and Q(5, 5, 10) internally in the ratio r : 1 (r > 0). If O is the origin and $(\vec{O Q} \cdot \vec{O A}) - \frac{1}{5} | \vec{O P} \times \vec{O A} |^{2} = 10$ , then the value of r is : [2025]

Smart Achievers · Accepted Answer

(1)

The point A divides line segment PQ internally in the ratio r : 1

$∴ A \equiv (\frac{5 r - 1}{r + 1}, \frac{5 r - 1}{r + 1}, \frac{10 r + 2}{r + 1})$

Now, $\vec{O Q} \cdot \vec{O A} = \frac{10}{r + 1} (15 r + 1)$

and $| \vec{O P} \times \vec{O A} |^{2} = \frac{r^{2}}{{(r + 1)}^{2}} (800)$

$(\vec{O Q} \cdot \vec{O A}) - \frac{| \vec{O P} \times \vec{O A} |^{2}}{5} = 10$

$\Rightarrow \frac{10}{r + 1} (15 r + 1) - \frac{1}{5} \frac{r^{2} (800)}{{(r + 1)}^{2}} = 10$

$\Rightarrow 2 r^{2} - 14 r = 0 \Rightarrow r = 7, r \neq 0$ .

Solution

Q.
Let the point A divide the line segment joining the points P(–1, –1, 2) and Q(5, 5, 10) internally in the ratio r : 1 (r > 0). If O is the origin and $(\vec{O Q} \cdot \vec{O A}) - \frac{1}{5} | \vec{O P} \times \vec{O A} |^{2} = 10$ , then the value of r is : [2025]

1 7

2 $\sqrt{7}$

3 3

4 14

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Solution

Q. Let the point A divide the line segment joining the points P(–1, –1, 2) and Q(5, 5, 10) internally in the ratio r : 1 (r > 0). If O is the origin and (OQ→·OA→)–15|OP→×OA→|2=10, then the value of r is : [2025]

1 7

2 7

3 3

4 14

Ans. (1) The point A divides line segment PQ internally in the ratio r : 1 ∴ A≡(5r–1r+1,5r–1r+1,10r+2r+1) Now, OQ→·OA→=10r+1(15r+1) and |OP→×OA→|2=r2(r+1)2(800) (OQ→·OA→)–|OP→×OA→|25=10 ⇒ 10r+1(15r+1)–15r2(800)(r+1)2=10 ⇒ 2r2–14r=0 ⇒ r=7, r≠0 .

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Q.
Let the point A divide the line segment joining the points P(–1, –1, 2) and Q(5, 5, 10) internally in the ratio r : 1 (r > 0). If O is the origin and $(\vec{O Q} \cdot \vec{O A}) - \frac{1}{5} | \vec{O P} \times \vec{O A} |^{2} = 10$ , then the value of r is : [2025]

2 $\sqrt{7}$