Question:
Let Q be the foot of perpendicular from the origin to the plane \(4x-3y+z+13=0\) and R be a point \((-1,-6)\) on the plane. Then length QR is :
Let Q be the foot of perpendicular from the origin to the plane \(4x-3y+z+13=0\) and R be a point \((-1,-6)\) on the plane. Then length QR is :
Show Hint
The point R must have three coordinates; the plane equation fixes the missing one. Then use \(QR^2=OR^2-OQ^2\) because \(OQ\perp QR\).
Updated On: Oct 31, 2026
- \(\sqrt{14}\)
- \(\sqrt{\frac{19}{2}}\)
- \(3\sqrt{\frac{7}{2}}\)
- \(\frac{3}{\sqrt2}\)
Show Solution
Verified By Collegedunia
The Correct Option is C
Solution and Explanation
The point R is printed with only two coordinates. The only way to complete it so that R lies on the plane and the result is one of the options is \(R=(-1,\,1,\,-6)\): check \(4(-1)-3(1)+(-6)+13=0\). Take this as the intended point.
Foot of perpendicular from the origin: \(Q=-\frac{13}{4^2+3^2+1^2}(4,-3,1)=-\frac12(4,-3,1)=\left(-2,\frac32,-\frac12\right)\).
\(OQ^2=\frac{13^2}{26}=\frac{13}{2}\), \(OR^2=1+1+36=38\).
Since \(OQ\perp QR\): \(QR^2=OR^2-OQ^2=38-\frac{13}{2}=\frac{63}{2}\).
\(QR=\sqrt{\frac{63}{2}}=3\sqrt{\frac72}\).
Answer: (3) \(3\sqrt{\frac{7}{2}}\)
Foot of perpendicular from the origin: \(Q=-\frac{13}{4^2+3^2+1^2}(4,-3,1)=-\frac12(4,-3,1)=\left(-2,\frac32,-\frac12\right)\).
\(OQ^2=\frac{13^2}{26}=\frac{13}{2}\), \(OR^2=1+1+36=38\).
Since \(OQ\perp QR\): \(QR^2=OR^2-OQ^2=38-\frac{13}{2}=\frac{63}{2}\).
\(QR=\sqrt{\frac{63}{2}}=3\sqrt{\frac72}\).
Answer: (3) \(3\sqrt{\frac{7}{2}}\)
Was this answer helpful?
0
0
Top JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
View More Questions
Top JEE Main 3D Geometry Questions
- Let line L pass through the point \( (0, 1, 2) \), intersect the line \( {2 = {3 = {4 \), and be parallel to the plane \( 2x + y - 3z = 4 \). Find the distance of the point \( P(1, -9, 2) \) from line L. \\
- Let the vectors \( , , \) represent three coterminous edges of a parallelepiped of volume \( V \). Then the volume of the parallelepiped, whose coterminous edges are represented by \( , + \) and \( + 2 + 3 \) is equal to: \\ \\
- If \( d_1 \) is the shortest distance between the lines \(\frac{x+1}{2} = \frac{y-1}{-12} = \frac{z-1}{-7} \quad \text{and} \quad \frac{x-1}{7} = \frac{y+8}{2} = \frac{z-4}{5},\) and \( d_2 \) is the shortest distance between the lines \(\frac{x-1}{2} = \frac{y-2}{1} = \frac{z-6}{-3} \quad \text{and} \quad \frac{x+2}{1} = \frac{y+2}{2} = \frac{z-1}{6},\) then the value of \( \frac{32\sqrt{3}d_1}{d_2} \) is:
- The lines \[ \frac{x - 2}{2} = \frac{y + 2}{-2} = \frac{z - 7}{16} \] and \[ \frac{x + 3}{4} = \frac{y + 2}{3} = \frac{z + 2}{1} \] intersect at the point \( P \). If the distance of \( P \) from the line \[ \frac{x + 1}{2} = \frac{y - 1}{3} = \frac{z - 1}{1} \] is \( l \), then \( 14l^2 \) is equal to \ldots
- Let PQR be a triangle with R (-1,4, 2). Suppose M(2, 1, 2) is the mid point of PQ. The distance of the centroid of \(\triangle PQR\) from the point of intersection of the line\(\frac{x-2}{0}=\frac{y}{2}=\frac{z+3}{-1}\) and \(\frac{x-1}{1}=\frac{y+3}{-3}=\frac{z+1}{1}\) is
View More Questions
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
View More Questions