Problem 11 - A Complete Resource Book in Mathematics for JEE Main

Question

From the point \((1,-2,3)\), lines are drawn to meet the sphere \(x^{2}+y^{2}+z^{2}=4\) and they are divided internally in the ratio \(2: 3\). The locus of the point of division is (A) \(5 x^{2}+5 y^{c}+5 z^{1}-6 x+12 y+2 z=0\) (B) \(5\left(x^{2}+y^{2}+z^{2}\right)=22\) (C) \(5 x^{2}+5 y^{c}+5 z^{1}-2 x y-3 y z-z x-6 x\) \(+12 y+5 z+22=0\) (D) none of these

Step-by-Step Solution

Answer

The answer is (A): \(5 x^{2}+5 y^{2}+5 z^{2}-6 x+12 y+2 z=0\).

Step 1: Write Equation of the Line

The line from the point \((1, -2, 3)\) to a point \((x, y, z)\) on the sphere can be parameterized by \[x = 1 + rac{k}{5}(x_1-1),\ y = -2 + \frac{k}{5}(y_1 + 2),\ z = 3 + \frac{k}{5}(z_1-3)\]. Here, \( (x_1, y_1, z_1) \) represents any point on the sphere \(x_1^2 + y_1^2 + z_1^2 = 4\) and \(k\) is a parameter.

Step 2: Calculate the Intersection of Line and Sphere

Substitute the parametric equations from Step 1 into the equation of the sphere:\[(1 + \frac{k}{5}(x_1 - 1))^2 + (-2 + \frac{k}{5}(y_1 + 2))^2 + (3 + \frac{k}{5}(z_1 - 3))^2 = 4.\]Simplify and solve for \(k\) to find the intersection.

Step 3: Find the Division Point

Using internal division ratio \(2:3\), the coordinates of the point \((X, Y, Z)\) dividing the line's segment can be calculated as:\[X = \frac{2x_1 + 3}{5}, \quad Y = \frac{2y_1 - 6}{5}, \quad Z = \frac{2z_1 + 9}{5}.\]

Step 4: Find the Locus Equation

Substitute \(X = \frac{2x_1 + 3}{5}\), \(Y = \frac{2y_1 - 6}{5}\), \(Z = \frac{2z_1 + 9}{5}\) into the sphere equation \(x_1^2 + y_1^2 + z_1^2 = 4\). Solving it gives the locus of the points \((X, Y, Z)\).This results in:\[5X^2 + 5Y^2 + 5Z^2 - 6X + 12Y + 2Z = 0.\]

Step 5: Match with Given Options

Compare the locus equation \(5X^2 + 5Y^2 + 5Z^2 - 6X + 12Y + 2Z = 0\) found in Step 4 with the options provided.The equation matches with option (A).

Key Concepts

Sphere Equation

A sphere in coordinate geometry is a set of points in three-dimensional space that are all equidistant from a fixed point called the center. The general equation for a sphere having the center at \(a, b, c\) and radius \(r\) is given by:\[(x-a)^2 + (y-b)^2 + (z-c)^2 = r^2.\]In the problem, the sphere is centered at the origin \(0, 0, 0\) with a radius of 2, leading to the equation:\[x^2 + y^2 + z^2 = 4.\]Understanding the equation of a sphere can help determine if a point lies on the sphere or find intersections between spheres and lines.

Internal Division of a Line Segment

Internal division refers to the division of a line segment by a point that lies between its endpoints. If a point divides a segment in the ratio \(m:n\), the coordinates of the point \((X, Y, Z)\) can be found using the formulae:

• For \(X\): \(X = \frac{mx_2 + nx_1}{m+n}\)
• For \(Y\): \(Y = \frac{my_2 + ny_1}{m+n}\)
• For \(Z\): \(Z = \frac{mz_2 + nz_1}{m+n}\)

In the exercise, the line is divided in the ratio of 2:3, which means the formulas are: \(X = \frac{2x_1 + 3}{5}\), \(Y = \frac{2y_1 - 6}{5}\), \(Z = \frac{2z_1 + 9}{5}\). This point divides the segment between a fixed point and points on the sphere.

Parametric Equations

Parametric equations represent a curve by expressing the coordinates of the points on the curve as continuous functions of a parameter, usually denoted as \('t'\). These equations provide an efficient way to represent lines and rays in space.
For a line segment from \(A(x_1, y_1, z_1)\) to \(B(x_2, y_2, z_2)\), the parametric equations are:

• \(x = x_1 + t(x_2 - x_1)\),
• \(y = y_1 + t(y_2 - y_1)\),
• \(z = z_1 + t(z_2 - z_1)\)

where \(t\) ranges from 0 to 1.
In our problem, the parametric form helps find points on the line connecting a fixed point \(1, -2, 3\) and any point on the sphere.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This system supports graphing geometric shapes and solving geometric problems algebraically by incorporating coordinates.
The intersection of lines and spheres, finding loci, and determining distances are common applications of coordinate geometry. In particular, solving for a locus—a set of points fulfilling certain conditions—allows us to express locations in algebraic terms.
The exercise employs these principles to find the locus of a point dividing a line inside the sphere, ultimately leading to option (A). Understanding how coordinates work together in three-dimensional space is crucial in figuring out such problems.