Question:
If N is the foot of the perpendicular drawn from the point \(P(5,-1,3)\) to the line passing through the points \(A(1,3,-5)\) and \(B(3,-1,5)\) then the ratio in which N divides AB is
If N is the foot of the perpendicular drawn from the point \(P(5,-1,3)\) to the line passing through the points \(A(1,3,-5)\) and \(B(3,-1,5)\) then the ratio in which N divides AB is
Show Hint
To find the foot of a perpendicular on a line in 3D, use the dot-product condition \((\overrightarrow{PN})\cdot(\overrightarrow{AB})=0\).
Updated On: Oct 17, 2026
- -26:21
- 13:2
- 11:19
- -11:19
Show Solution
Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Concept:
The foot of the perpendicular from a point to a line lies on the line itself. If \[ N=A+t(B-A) \] then \(\overrightarrow{PN}\) must be perpendicular to \(\overrightarrow{AB}\).
Step 1: Find the direction vector of line AB. \[ \overrightarrow{AB} =(3-1,\,-1-3,\;5-(-5)) \] \[ =(2,-4,10) \]
Step 2: Write the coordinates of \(N\). \[ N=(1,3,-5)+t(2,-4,10) \] \[ N=(1+2t,\;3-4t,\;-5+10t) \]
Step 3: Form the vector \(\overrightarrow{PN}\). \[ \overrightarrow{PN} =(1+2t-5,\;3-4t+1,\;-5+10t-3) \] \[ =(2t-4,\;4-4t,\;10t-8) \]
Step 4: Apply perpendicularity condition. \[ \overrightarrow{PN}\cdot\overrightarrow{AB}=0 \] \[ 2(2t-4)-4(4-4t)+10(10t-8)=0 \] \[ 4t-8-16+16t+100t-80=0 \] \[ 120t-104=0 \] \[ t=\frac{104}{120} =\frac{13}{15} \]
Step 5: Find the division ratio. Since \[ AN:NB=t:(1-t) \] \[ =\frac{13}{15}:\frac{2}{15} \] \[ =13:2 \] \[ \boxed{13:2} \]
The foot of the perpendicular from a point to a line lies on the line itself. If \[ N=A+t(B-A) \] then \(\overrightarrow{PN}\) must be perpendicular to \(\overrightarrow{AB}\).
Step 1: Find the direction vector of line AB. \[ \overrightarrow{AB} =(3-1,\,-1-3,\;5-(-5)) \] \[ =(2,-4,10) \]
Step 2: Write the coordinates of \(N\). \[ N=(1,3,-5)+t(2,-4,10) \] \[ N=(1+2t,\;3-4t,\;-5+10t) \]
Step 3: Form the vector \(\overrightarrow{PN}\). \[ \overrightarrow{PN} =(1+2t-5,\;3-4t+1,\;-5+10t-3) \] \[ =(2t-4,\;4-4t,\;10t-8) \]
Step 4: Apply perpendicularity condition. \[ \overrightarrow{PN}\cdot\overrightarrow{AB}=0 \] \[ 2(2t-4)-4(4-4t)+10(10t-8)=0 \] \[ 4t-8-16+16t+100t-80=0 \] \[ 120t-104=0 \] \[ t=\frac{104}{120} =\frac{13}{15} \]
Step 5: Find the division ratio. Since \[ AN:NB=t:(1-t) \] \[ =\frac{13}{15}:\frac{2}{15} \] \[ =13:2 \] \[ \boxed{13:2} \]
Was this answer helpful?
0
0
Top AP EAPCET Mathematics Questions
- $D = \left\{x\in\mathbb{R}: f\left(x\right) =\sqrt{\frac{x - \left|x\right|}{x - \left[x\right]}} \text{is defined} \right\}$ and $C$ be the range of the real function $g(x) = \frac{2x}{4 + x^{2}}$. Then $D \cap C$
- $f\left(x\right) = \frac{x}{e^{x}-1} + \frac{x}{2} + 2 \cos^{3} \frac{x}{2} $ on $R-\left\{0\right\} $ is
- Consider the following lists.
List-I List-II (A) $f(x) = \frac{|x+2|}{x+2} , x \ne -2 $ (I) $[\frac{1}{3} , 1 ]$ (B) $(x)=|[x]|,x \in [R$ (II) Z (C) $h(x) = |x - [x]| , x \in [R$ (III) W (D) $f(x) = \frac{1}{2 - \sin 3x} , x \in [R$ (IV) [0, 1) (V) { -1, 1} - If $[x]$ is the greatest integer less than or equal to $x$ and $|x|$ is the modulus of $x$. then the system of three equations $2x + 3 | y | + 5[z] = 0, x + |y| - 2[z] = 4, x + |y| + |z| = 1$ has
- Investigate the values of $\lambda$ and $\mu$ for the system $x+2 y+3 z=6, x+3 y+5 z=9$, $2 x+5 y+\lambda z=\mu$ and match the values in List - I with the items in List - II.
List I List II (A) $\lambda=8, \mu \neq 15$ 1. Infinitely many solutions (B) $\lambda \neq 8, \mu \in R$ 2. No solution (C) $\lambda=8, \mu=15$ 3. Unique solution
View More Questions
Top AP EAPCET 3D Geometry Questions
- If the line with direction ratios \[ (1, a, \beta) \] is perpendicular to the line with direction ratios \[ (-1,2,1) \] and parallel to the line with direction ratios \[ (\alpha,1,\beta), \] then \( (\alpha, \beta) \) is:
- The equation of the plane which bisects the angle between the planes $$ 3x - 6y + 2z + 5 = 0 \quad \text{and} \quad 4x - 12y + 3z - 3 = 0 \quad \text{which contains the origin is:} $$
- The distance of the point \( O(0,0,0) \) from the plane \( \overrightarrow{r} \cdot (\hat{i} + \hat{j} + \hat{k}) = 5 \) measured parallel to \( 2\hat{i} + 3\hat{j} - 6\hat{k} \) is?
If \( A(1,0,2) \), \( B(2,1,0) \), \( C(2,-5,3) \), and \( D(0,3,2) \) are four points and the point of intersection of the lines \( AB \) and \( CD \) is \( P(a,b,c) \), then \( a + b + c = ? \)
The distance from a point \( (1,1,1) \) to a variable plane \(\pi\) is 12 units and the points of intersections of the plane with X, Y, Z-axes are \( A, B, C \) respectively. If the point of intersection of the planes through the points \( A, B, C \) and parallel to the coordinate planes is \( P \), then the equation of the locus of \( P \) is:
View More Questions
Top AP EAPCET Questions
- A body is projected from the ground at an angle of $\tan^{-1} \left( \frac{8}{7}\right)$ with the horizontal. The ratio of the maximum height attained by it to its range is
- The four quantum numbers of the valence electron of potassium are :
- A lens forms real and virtual images of an object, when the object is at $u_{1}$ and $u_{2}$ distances respectively. If the size of the virtual image is double that of the real image, then the focal length of the lens is (take, the magnification of the real image as $m$ )
- Two spheres $P$ and $Q$, each of mass $200\, g$ are attached to a string of length one metre as shown in the figure. The string and the spheres are then whirled in a horizontal circle about $O$ at a constant angular speed. The ratio of the tension in the string between $P$ and $Q$ to that of between $P$ and $O$ is $(P$ is at mid-point of the line joining $O$ and $Q$ )
- The volume of $0.02\, M$ acidified permanganate solution required for complete reaction of $60\, mL$ of $0.01\, MI^{-}$ ion solution to form $I_2$ in $mL$ is
View More Questions