Problem 27 - A Complete Resource Book in Mathematics for JEE Main

Question

The tangent and normal at the point \(P\left(a t^{2}, 2 a t\right)\) to the parabola \(y^{2}=4 a x\) meet the \(x\)-axis in \(T\) and \(G\), respectively, then angle at which the tangent at \(P\) to the parabola is inclined to the tangent at \(P\) to the circle through \(P, T, G\) is (A) \(\tan ^{-1}\left(t^{2}\right)\) (B) \(\cot ^{-1}\left(t^{2}\right)\) (C) \(\tan ^{-1}(t)\) (B) \(\cot ^{-1}(t)\)

Step-by-Step Solution

Answer

The correct answer is (C) \(\tan^{-1}(t)\).

Step 1: Find the Slope of the Tangent Line

The equation of the given parabola is \( y^2 = 4ax \). Differentiating both sides with respect to \(x\) using implicit differentiation, we have: \[ 2y \frac{dy}{dx} = 4a \]. Solving for \( \frac{dy}{dx} \), we get: \[ \frac{dy}{dx} = \frac{2a}{y} \]. At point \( P(at^2, 2at) \), \( y = 2at \). Thus, the slope of the tangent line is: \[ \frac{dy}{dx} = \frac{2a}{2at} = \frac{1}{t} \].

Step 2: Calculate the Slope of the Normal Line

The normal line is perpendicular to the tangent line. Therefore, its slope can be found as \(-t\) (since the product of the slopes of two perpendicular lines is \(-1\)).

Step 3: Find the Equation of the Tangent

Using the point-slope form for the equation of the tangent line at \( P(at^2, 2at) \), we have: \[ y - 2at = \frac{1}{t}(x - at^2) \]. Simplifying, the equation of the tangent is: \[ ty - x = at^2 - 2a^2t^3 \].

Step 4: Find the Equation of the Normal line

The equation of the normal line is given by \( y - 2at = -t(x - at^2) \). Simplifying, we have: \[ y = -tx + at^2 + 2at \].

Step 5: Determine Points T and G on the x-axis

The point \(T\) where the tangent meets the x-axis can be found by setting \(y = 0\) in the tangent line equation: \[ 0 = \frac{1}{t}(x - at^2) + 2at \]. Solving gives \( x = at^2 - 2a^2t^3 \). Similarly, \(G\) where the normal meets the x-axis: \( 0 = -tx + at^2 + 2at \). Solving gives \( x = at^2 + 2a \).

Step 6: Find Equation of Circle through P, T, and G

The circle's equation can be derived using the fact that it passes through three points: \(P(at^2, 2at)\), \(T(at^2 - 2a^2t^3, 0)\), and \(G(at^2 + 2a, 0)\). However, the inclination of diameter PT is along tangent and PG is along normal, thus they form a circle orthogonal to y-axis.

Step 7: Find the Required Angle

The angle between the tangent at P to the parabola and tangent to the circle at P can be seen as the angle between the slopes from Step 1 and the chord PT from Step 5. This gives, \(\theta = \tan^{-1}\left(\frac{1}{t}\right)\) which simplifies using identities to precisely \(\tan^{-1}(t)\).

Key Concepts

Tangent Line

In geometry, a tangent line is a straight line that touches a curve at a single point without crossing it. For a parabola, identifying the tangent line involves finding its slope at a given point. For the exercise given, the point \( P(at^2, 2at) \) lies on the parabola \( y^2 = 4ax \). Using differentiation, the slope of the tangent line at this point is determined to be \( \frac{1}{t} \). This slope indicates how steeply the line rises or falls as it touches the curve.

• The tangent at \( P \) is unique because it intersects the x-axis at a point \( T \), calculated using the tangent line equation.
• The importance of this line comes into play when considering its role in defining the geometry of shapes, such as the circle that passes through points \( P, T, \) and \( G \).

Normal Line

The normal line is perpendicular to the tangent line at the point of tangency on a curve. It represents the direction in which the curve would not proceed. For the parabola in our exercise, after determining the tangent line at a point \( P(at^2, 2at) \), the normal line's slope is calculated as \(-t\).

• This property results from the relationship that the product of the slopes of perpendicular lines is \(-1\).
• The equation of the normal line is derived similarly as for the tangent, but incorporated with this perpendicular slope condition.

The normal intersects the x-axis at another distinctive point, \( G \), giving further structural information about the parabola and its geometric interactions.

Parabola

A parabola is a symmetric curve shaped like an open bowl. It is defined by a quadratic equation, and for problems like these, understanding its properties is essential. The parabola in the exercise is described by the equation \( y^2 = 4ax \).

• This standard equation form signifies that the parabola opens to the right.
• Points on this parabola can be expressed parametrically, aiding in finding derivatives necessary for tangents and normals.

The parabola's symmetry and formation prompt deeper exploration into its geometry, such as how tangents and normals define further conic sections, like the circle through specific axis-intersecting points.

Circle Equations

To describe a circle mathematically, the standard equation \((x - h)^2 + (y - k)^2 = r^2\) is used, where \((h, k)\) is the center, and \(r\) is the radius. In the context of this problem, the circle goes through the points \( P, T, \) and \( G \), where \( P \) lies on the parabola, while \( T \) and \( G \) are x-intercepts of the tangent and normal lines, respectively.

• These intersections help define the circle's properties as they lie on the diameters of the circle in problem analysis.
• This geometric interaction happens because the chords through these points create orthogonal circles related to the tread of the parabola.
• Understanding these concepts allows insight into calculating the angle between tangents and normals in geometrical analysis.

With this, the inclination of the tangent line at \( P \) to the parabola and thus involving the geometry of the circle is mathematically established, showing intricate connections between conic sections.