Question

A line passing through \(0(0,0)\) intersect the parallel lines \(4 x+2 y=0\) and \(2 x+y+6=0\) at \(P\) and \(Q\) respectively, then in what ratio does 0 divide \(\underline{\text { PQ from }} \mathrm{P}\) ? (a) \(1: 2\) (b) \(3: 4\) (c) \(2: 1\) (d) \(4: 3\)

Short answer

Please recheck the given exercise and ensure all the given information is accurate.

Step-by-step solution

Finding Point P

To find the point P, we need to solve the equations of the given line and the equation of the first parallel line. Let the given line be L1: y = m1x Given parallel line: \[4x + 2y = 0\] Substituting L1 into the parallel line's equation, we get: \[4x + 2(m1x) = 0\] \[(4 + 2m1)x = 0\] Since this is not the trivial case, we can say that x = 0 Now, substituting x = 0 into the equation for L1, we get y = 0. So point P is (0,0).

Finding Point Q

To find the point Q, we need to solve the equations of the given line and the equation of the second parallel line. Given line L1: y = m1x Given parallel line: \[2x + y + 6 = 0\] Substituting L1 into the second parallel line's equation, we get: \[2x + m1x + 6 = 0\] \[(2 + m1)x = -6\] Since this is not the trivial case, we can say that x ≠ 0 and m1 ≠ -2. Solving this, we get x = -6 / (2 + m1) and y = m1 * (-6 / (2 + m1)) So point Q is (-6 / (2 + m1), m1 * (-6 / (2 + m1)))

Applying the Section Formula

Now, applying the section formula to find the ratio in which O divides PQ. Let's define the ratio as k:1 x-coordinate: \[0 = \frac{k(-6 / (2 + m1)) + 1 * 0}{k + 1}\] Since -6 / (2 + m1) ≠ 0, we can say that k = 0. y-coordinate: \[0 = \frac{k(m1 * (-6 / (2 + m1))) + 1 * 0}{k + 1}\] Since k = 0, we only need to solve y-coordinate equation: \[0 = m1 * (-6 / (2 + m1))\] Since m1 ≠ 0, we can say that -6 / (2 + m1) = 0 and m1 = -2 However, our previous assumption about m1 ≠ -2 was wrong. The problem statement seems to have some inconsistency. Please recheck the given exercise and ensure all the given information is accurate.

Key concepts

Coordinate Geometry

In the realm of JEE Mathematics, coordinate geometry is a fascinating subject that merges algebra with geometry to solve problems involving positions and shapes. The basic idea is to represent points on a plane using pairs of numbers, typically denoted as \(x, y\). These numbers correspond to the point's horizontal and vertical distances from a reference point, known as the origin (0,0).

Coordinate geometry allows us to create equations that describe geometric figures like lines, circles, and polygons. Lines, for example, can be expressed using various forms of linear equations, such as slope-intercept form \(y = mx + c\), where \(m\) represents the slope and \(c\) the y-intercept.

• The concept of slopes and intercepts helps in understanding how lines can be parallel, perpendicular, or intersect.
• Distances between points on a plane can be calculated, adding an analytical approach to geometry.

Mastery of coordinate geometry can significantly ease solving complex geometric problems with precision and clarity.

Section Formula

The section formula is a crucial concept for dividing a line into a certain ratio. Imagine a line segment \(PQ\) with endpoints \(P(x_1, y_1)\) and \(Q(x_2, y_2)\). This formula helps in finding a point \(R(x, y)\) that divides the line segment in a given ratio \(m:n\).

The section formula states that the coordinates of \(R\) are given by the equations:

• For x-coordinate: \(x = \frac{mx_2 + nx_1}{m+n}\)
• For y-coordinate: \(y = \frac{my_2 + ny_1}{m+n}\)

When the division is internal, meaning \(R\) is between \(P\) and \(Q\), the section formula applies directly.

This formula is not only useful in geometry but is also often employed in physics and engineering to solve problems involving division of distances, mass, or even electric charges. It elegantly translates the geometric understanding of division into algebraic expressions.

Parallel Lines

Parallel lines are a fundamental concept in both geometry and algebra. In the context of coordinate geometry, parallel lines lie in the same plane and never meet, no matter how far they extend. This occurs because they have the same slope.

For instance, two lines with equations \(y = m_1x + b_1\) and \(y = m_2x + b_2\) are parallel if \(m_1 = m_2\). It's their identical slopes that maintain the consistent, parallel distance.

• Parallel lines are often used in problem-solving as they simplify calculations by maintaining equal slopes.
• In distances and intersections, knowing lines are parallel can considerably reduce computational complexity.

In solving equations involving parallel lines, concepts like determinant zero-condition are also applied, confirming that no common points exist on each line. Recognizing parallelism is key in proving geometric hypotheses and can ease the journey through complex problem spaces.

Intersection Point Calculation

Calculating the intersection point of two lines is an essential skill in geometry. This process involves finding the exact point where two lines meet when given their equations. To find this, you set the equations of the two lines equal and solve for the coordinates \(x, y\).

Consider two lines \(L1: y = m_1x + c_1 \) and \(L2: y = m_2x + c_2 \). To find their intersection, equate: \(m_1x + c_1 = m_2x + c_2 \). Solving this gives the x-coordinate at their intersection:

• Simplify to find x: \(x = \frac{c_2-c_1}{m_1-m_2} \)

Using this x-value in one of the line's equations will give the y-coordinate.

This method is straightforward when lines are not parallel, as non-parallel lines will always intersect at one point. Mastery of intersection point calculation is vital for more advanced topics like the intersection of curves or solving systems of equations. Recognizing and applying this skill reduces complexity in grasping multiple geometrical problems.