The nearest point on the line $3x + 4y = 12$ from the origin is
- $\left(\frac{36}{25} , \frac{48}{25}\right) $
- $\left(3 , \frac{3}{4}\right) $
- $\left(2 , \frac{3}{2}\right) $
- None of these
The Correct Option is A
Solution and Explanation
If ‘D’ be the foot of altitude, drawn from origin to the given line, then �$D$� is the required point.
Let $\angle OBA = \theta$
$\Rightarrow \, \tan \theta = 4/3$
$\Rightarrow \, \angle DOA = \theta $
we have $OD = 12/5$
If $D$ is $(h, k)$ then $h = OD \cos \theta, k = OD \sin \theta$
$\Rightarrow \, h = 36/25, k = 48/25$.
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Concepts Used:
Distance of a Point From a Line
The length of the perpendicular drawn from the point to the line is the distance of a point from a line. The shortest difference between a point and a line is the distance between them. To move a point on the line it measures the minimum distance or length required.
To Find the Distance Between two points:
The following steps can be used to calculate the distance between two points using the given coordinates:
- A(m1,n1) and B(m2,n2) are the coordinates of the two given points in the coordinate plane.
- The distance formula for the calculation of the distance between the two points is, d = √(m2 - m1)2 + (n2 - n1)2
- Finally, the given solution will be expressed in proper units.
Note: If the two points are in a 3D plane, we can use the 3D distance formula, d = √(m2 - m1)2 + (n2 - n1)2 + (o2 - o1)2.
Read More: Distance Formula