MHT CET · Maths · Three Dimensional Geometry
The length of the perpendicular from the point \((0,2,3)\) on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}\)
- A \(\sqrt{15}\) units
- B \(\sqrt{21}\) units
- C \(\sqrt{33}\) units
- D \(\sqrt{11}\) units
Answer & Solution
Correct Answer
(B) \(\sqrt{21}\) units
Step-by-step Solution
Detailed explanation
Any point on the line \(\frac{x+3}{5}=\frac{y-1}{2}=\frac{z+4}{3}=\lambda\) can be taken as
\(
(5 \lambda-3,2 \lambda+1,3 \lambda-4)
\)
\(\text {for foot of perpendicular } \lambda=\) \(\frac{a\left(\alpha-x_1\right)+b\left(\beta-y_1\right)+c\left(\gamma-z_1\right)}{a^2+b^2+c^2} \)
\( =\frac{5(0+3)+2(2-1)+3(3+4)}{5^2+2^2+3^2} \)
\( \text {i.e., } \lambda=1 \)
\( \Rightarrow \text {foot of perpendicular }(2,3,-1) \)
\( \Rightarrow \text {Length of the perpendicular }=\) \(\sqrt{(0-2)^2+(2-3)^2+(3+1)^2} \)
\( =\sqrt{21}\)
\(
(5 \lambda-3,2 \lambda+1,3 \lambda-4)
\)
\(\text {for foot of perpendicular } \lambda=\) \(\frac{a\left(\alpha-x_1\right)+b\left(\beta-y_1\right)+c\left(\gamma-z_1\right)}{a^2+b^2+c^2} \)
\( =\frac{5(0+3)+2(2-1)+3(3+4)}{5^2+2^2+3^2} \)
\( \text {i.e., } \lambda=1 \)
\( \Rightarrow \text {foot of perpendicular }(2,3,-1) \)
\( \Rightarrow \text {Length of the perpendicular }=\) \(\sqrt{(0-2)^2+(2-3)^2+(3+1)^2} \)
\( =\sqrt{21}\)
See the Complete Solution
Get step-by-step explanations for this and 2.5 Lakh+ more JEE, NEET & CET questions.
- Unlock all solutions
- Practice the full chapter
- Track accuracy across PYQs
4.8 rated on Google Play · 14,000+ reviews
More questions from Maths
- Two circles centered at \((2,3)\) and \((4,5)\) intersects each other. If their radii are equal, then the equation of the common chord isMHT CET 2021 Medium
- The value of \(\sin 18^{\circ}\) isMHT CET 2019 Medium
- If \(3 \sin 2 \theta=2 \sin 3 \theta\) and \(0 < \theta < \pi\), then the value of \(\sin \theta\) is equal toMHT CET 2025 Medium
- The co-ordinates of point on the line \(x+y+3=0\), whose distance from the line \(x+2 y+2=0\) is \(\sqrt{5}\) units, areMHT CET 2022 Easy
- A random variable \(X\) has following p.d.f.
\(\mathrm{f}(\mathrm{x})=\mathrm{k} x(1-x), 0 \leqslant x \leqslant 1\)
and \(\mathrm{P}(x>a)=\frac{20}{27}\), then \(a=\)MHT CET 2025 Medium - The approximate value of \(\cot ^{-1}(1 \cdot 001)\) isMHT CET 2020 Easy
More PYQs from MHT CET
- If \(\mathrm{x}=\mathrm{e}^{\mathrm{t}}(\sin \mathrm{t}-\cos \mathrm{t})\) and \(\mathrm{y}=\mathrm{e}^{\mathrm{t}}(\sin \mathrm{t}+\cos \mathrm{t})\), then \(\frac{\mathrm{dy}}{\mathrm{dx}}\) at \(\mathrm{t}=\frac{\pi}{3}\) isMHT CET 2021 Medium
- If \(x=\cos ^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right), y=\sin ^{-1}\left(\frac{1}{\sqrt{1+t^2}}\right)\), then \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) isMHT CET 2022 Hard
- \(\int \frac{d x}{\sqrt{5+4 x-x^{2}}}=\)MHT CET 2020 Easy
- The curve \(\left(\frac{x}{a}\right)^n+\left(\frac{y}{b}\right)^n=2, n \in N\) touches the line at the point \((a, b)\), then the equation of the line isMHT CET 2022 Hard
- The molar specific heat of an ideal gas at constant pressure and constant volume is \(\mathrm{C}_{\mathrm{p}}\) and \(\mathrm{C}_{\mathrm{v}}\) respectively. If \(\mathrm{R}\) is universal gas constant and \(\gamma=\frac{C_p}{C_y}\) then \(C_v=\)MHT CET 2023 Medium
- A thin wire of length ' \(L\) ' and uniform linear mass density ' \(\mathrm{m}\) ' is bent into a circular coil. The moment of inertia of this coil about tangential axis and in plane of the coil isMHT CET 2023 Hard