Problem 28 - A Complete Resource Book in Mathematics for JEE Main

Question

If normals are drawn from a point \(P(h, k)\) to the parabola \(y^{2}=4 a x\), then the sum of the intercepts which the normals cut off from the axis of the parabola is (A) \((h+a)\) (B) \(3(h+a)\) (C) \(2(h+a)\) (D) none of these

Step-by-Step Solution

Answer

The sum of the intercepts is \(3(h + a)\).

Step 1: Standard form of the parabola

The given parabola is in the form \(y^2 = 4ax\). For this standard form, the parabola opens to the right with the vertex at the origin \((0,0)\).

Step 2: Formula for normal to the parabola

The equation of the normal at any point \((at^2, 2at)\) on the parabola \(y^2 = 4ax\) is given by: \ t^3y + (t^2 - 2a)x = 2at^3 + 2at.

Step 3: Substituting point P into normal equation

Since normals are drawn from the point \(P(h, k)\), we substitute \((h, k)\) into the normal equation: \ t^3k + (t^2 - 2a)h = 2at^3 + 2at.

Step 4: Solving for t using cubic equation

By rearranging and solving the equation from Step 3 for \(t\), we get a cubic equation: \ t^3h - 2at^3 + t^2k - 2at + ht^2 = 0. This represents the values of \(t\) for which the given normals are valid.

Step 5: Sum of the intercepts on the axis

By Vieta's formulas and properties of cubic equations, the sum of the roots of the equation in step 4, which are the slopes at which norms cut the axis, gives: \ \(-\frac{coefficient\ of\ (t^2)}{coefficient\ of\ (t^3)} = -(\frac{h+k}{h})\). We also know each intercept to be \((h+a)\).

Step 6: Final calculation of intercepts sum

Since each intercept is \((h + a)\) and there are 3 such intercepts (from the properties of cubic equations), the sum of intercepts is \ 3(h+a).

Key Concepts

Normal to Parabola

When we talk about the normal to a parabola, we are looking at lines that are perpendicular to the tangent of the parabola at a certain point. The parabola in question is defined by the equation \( y^2 = 4ax \). This parabola opens to the right, and its vertex is located at the origin \((0, 0)\). For any point \((at^2, 2at)\) on this parabola, a normal can be drawn, which creates a perpendicular line at that point. To find the equation of the normal to the parabola, we rely on the expression:- \( t^3y + (t^2 - 2a)x = 2at^3 + 2at \)
This formula helps us derive the relationship necessary to analyze the points where the normals intersect the parabola's axis. Being familiar with this process allows for a deeper understanding of the geometry of conic sections. It also develops a foundation that's crucial when delving into more advanced parabola problems that involve normals.

Intercepts

Intercepts are the points where a line crosses the axes of a coordinate plane. When discussing the intercepts concerning a parabola, we often refer to these points as vital information about lines such as the normal. For a parabola with a given equation, the intercepts can be calculated by plugging specific values into our normal equation. These calculations often require us to solve complex expressions to find the lines' intersection with the axis.In our particular problem, the exercise involves finding the sum of these intercepts when normals are drawn from a specific point \((h, k)\). Understanding this concept helps us bridge algebraic solutions and geometrical interpretations, which is a valuable skill in many areas of mathematics.
By knowing how to find intercepts, you'll be better equipped to understand the overall shape and position of lines or curves on a plane.

Cubic Equation

A cubic equation is an equation of the form \( ax^3 + bx^2 + cx + d = 0 \). In our exercise, we arrive at a cubic equation when trying to solve for the parameter \( t \) from the normal equation substituting the point \((h, k)\). The cubic equation is given as:- \( t^3h - 2at^3 + t^2k - 2at + ht^2 = 0 \)
Solving this equation gives us the values of \( t \), which are tied to the slopes of our normals. When dealing with cubic equations, the solutions or "roots" provide critical information about intersections, behavior of functions, and real-world mathematical modeling. While solving cubic equations might seem challenging, understanding their structure and properties opens up a vast array of mathematical insights for problems involving conics like parabolas.

Vieta's Formulas

Vieta's formulas are powerful tools that relate the coefficients of polynomials to sums and products of their roots. For a cubic equation like \( ax^3 + bx^2 + cx + d = 0 \), Vieta's formulas tell us that:

• The sum of the roots \( r_1 + r_2 + r_3 = -\frac{b}{a} \)
• The sum of the product of the roots taken two at a time \( r_1r_2 + r_2r_3 + r_3r_1 = \frac{c}{a} \)
• The product of the roots \( r_1r_2r_3 = -\frac{d}{a} \)

Applying Vieta's formulas in our exercise involves using the specific coefficients of the constructed cubic equation for \( t \). From these, we glean that the sum of the intercepts corresponds to the negated coefficient of the \( t^2 \) term divided by the coefficient of the \( t^3 \) term. This gives a clear and calculating path to solving such intercept problems quickly.
Understanding and applying Vieta's formulas make solving polynomial-related problems much more manageable. They offer significant shortcuts in mathematical problem-solving, which is why they're such a favored tool among mathematicians.