The line of the shortest distance between the lines \(\frac{x-2}{0}=\frac{y-1}{1}=\frac{z}{1}\) and \(\frac{x-3}{2}=\frac{y-5}{2}=\frac{z-1}{1}\) makes an angle of cos−1\(\sqrt{\frac{2}{27}}\) with the plane P: ax – y – z = 0, (a> 0). If the image of the point (1, 1, –5) in the plane P is (α, β, γ), then α + β – γ is equal to ________.
Correct Answer: -2.2 - -2.3
Solution and Explanation
The line of shortest distance will be along \(\vec {b_1}\)×\(\vec{b_2}\)
Where,
\(\vec {b_1}\)=\(\hat j\)+\(\hat k\)
and
\(\vec {b_2}\)=2\(\hat i\)+2\(\hat j\)+\(\hat k\)
\(\vec {b_1}\)×\(\vec {b_2}\)=\(\begin{vmatrix} \hat i &\hat j &\hat k \\ 0&0 &1 \\ 2&2 &1 \end{vmatrix}\)=−\(\hat i\)+2\(\hat j\)−2\(\hat k\)
Angle between \(\vec {b_1}\)×\(\vec {b_2}\) and plane P,
sinθ=|\(\frac{-a-2+2}{3.\sqrt{a^2+2}}\)|=\(\frac{5}{\sqrt{27}}\)
⇒\(\frac{|a|}{\sqrt{a^2+2}}\)=\(\frac{5}{\sqrt3}\)
⇒ a2=-\(\frac{25}{11}\)(not possible)
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_____
Top JEE Main distance between two points Questions
- \(L_1: γ^-=(i+2j+3k)+λ(i-j+k)\);
\(L_2: γ^-=(4i+5j+6k)+μ(i+j-k)\),
intersect \(L_1\) and \(L_2\) at P and Q respectively. If \((α, β, γ)\) is the mid point of the line segment PQ, then \(2(α, β, γ\)) is equal to - There are 20 lines numbered as 1,2,3,..., 20. And the odd numbered lines intersect at a point and all the even numbered lines are parallel. Find the maximum number of point of intersections
- Let the line passing through the points \( (-1, 2, 1) \) and parallel to the line \[ \frac{x - 1}{2} = \frac{y + 1}{3} = \frac{z}{4} \] intersect the line \[ \frac{x + 2}{3} = \frac{y - 3}{2} = \frac{z - 4}{1} \] at the point P. Then the distance of P from the point Q(4, -5, 1) is:
Let the plane P pass through the intersection of the planes \(2 x+3 y-z=2\)and \(x+2 y+3 z=6,\) and be perpendicular to the plane \(2 x+y-z+1=0\)If d is the distance of P from the point (-7,1,1), then \(d^2\) is equal to :
- Let AB be a chord of length 12 of the circle
\(\begin{array}{l}(x-2)^2 + (y+1)^2=\frac{169}{4}.\end{array}\)
If tangents drawn to the circle at points A and B intersect at the point P, then five times the distance of point P from chord AB is equal to _____ .
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,
Concepts Used:
Shortest Distance Between Two Parallel Lines
Formula to find distance between two parallel line:
Consider two parallel lines are shown in the following form :
\(y = mx + c_1\) …(i)
\(y = mx + c_2\) ….(ii)
Here, m = slope of line
Then, the formula for shortest distance can be written as given below:
\(d= \frac{|c_2-c_1|}{\sqrt{1+m^2}}\)
If the equations of two parallel lines are demonstrated in the following way :
\(ax + by + d_1 = 0\)
\(ax + by + d_2 = 0\)
then there is a little change in the formula.
\(d= \frac{|d_2-d_1|}{\sqrt{a^2+b^2}}\)