Problem 93 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The centre of the circle passing through the point \((0,1)\) and touching the parabola \(y=x^{2}\) at the point \((2,4)\) is: (a) \(\left(\frac{-53}{10}, \frac{16}{5}\right)\) (b) \(\left(\frac{6}{5}, \frac{53}{10}\right)\) (c) \(\left(\frac{3}{10}, \frac{16}{5}\right)\) (d) \(\left(\frac{-16}{5}, \frac{53}{10}\right)\)

Step-by-Step Solution

Answer

The center is \((-\frac{16}{5}, \frac{53}{10})\). Option (d).

Step 1: Define Centre of Circle

Assume the centre of the circle is \((h, k)\). The radius of the circle is \(r\).

Step 2: Circle Equation

The general equation of a circle with centre \((h, k)\) is given by:\[(x - h)^2 + (y - k)^2 = r^2\]Since it passes through the point \((2, 4)\), substitute \(x = 2\) and \(y = 4\):\[(2 - h)^2 + (4 - k)^2 = r^2\]

Step 3: Condition for Touching the Parabola

Since the circle touches the parabola \(y = x^2\) at \(x = 2\), the curve \(y = x^2\) at \((2, 4)\) has the slope \(m = 2x = 4\). The radius of the circle at this point is perpendicular to this slope, giving:\[m_{circle} \times m_{parabola} = -1\]

Step 4: Equation for Tangent Condition

Using the perpendicular condition, \(\frac{k - 4}{h - 2} = -\frac{1}{4}\). Rearrange:\[k - 4 = -\frac{1}{4}(h - 2)\]Solve for \(k\):\[k = -\frac{1}{4}h + \frac{9}{2}\]

Step 5: Circle Passes Through (0,1)

The circle also passes through the point \((0, 1)\), leading to another equation:\[(h - 0)^2 + (k - 1)^2 = r^2\]

Step 6: Solve System of Equations

Now we have two equations:1. \((2 - h)^2 + (4 - k)^2 = r^2\) 2. \(h^2 + (k - 1)^2 = r^2\)Replace \(k\) from Step 4 into both equations to find \(h\) and then use that to find \(k\).

Step 7: Calculate Centre Coordinates

Solve the simultaneous equations using substitution method:\[h \approx -\frac{16}{5}, \, k \approx \frac{53}{10}\]

Step 8: Verify Solution

Verify that substituting \(h = -\frac{16}{5}\) and \(k = \frac{53}{10}\) into the original equations of the circle satisfy both conditions.

Key Concepts

Circle Equation

The equation of a circle is an essential concept in coordinate geometry and can easily describe any circle using its center and radius. The standard form of a circle's equation is: \[(x - h)^2 + (y - k)^2 = r^2\] Here,

• \( (h, k) \) represents the coordinates of the circle's center.
• \( r \) is the radius.

This equation allows us to understand that any adjustments in \( h \) or \( k \) translate the circle along the x-axis or y-axis respectively. This form is also useful when given points through which the circle passes, as you can substitute these points to derive the necessary parameters of the circle like in the solution provided.

Parabola

A parabola is a unique curve represented by a specific quadratic equation, and in coordinate geometry, it commonly takes the form \( y = ax^2 + bx + c \). Often, parabolas are oriented with their vertex at the origin, simplifying to \( y = x^2 \). Key features include:

• The vertex, which is the highest or lowest point on the curve.
• The axis of symmetry, a vertical line that passes through the vertex.
• The direction in which the parabola opens, determined by the sign of \( a \).

In this exercise, we're examining a parabola that opens upwards, described by \( y = x^2 \). Understanding the vertex and directrix enables us to solve complex problems involving intersections and tangents with other figures, like circles.

Coordinate Geometry

Coordinate geometry, or analytic geometry, is a branch of geometry where the position of points on the plane is given by coordinates. This powerful tool combines algebra and geometry to solve problems involving geometric figures. Key elements of coordinate geometry include:

• Points, represented as \( (x, y) \).
• Line equations, such as \( y = mx + c \), describing the slope-intercept form of a straight line.
• Curves, such as circles and parabolas, defined by specific equations.

By applying algebraic techniques, we can find intersections, distances, and angles between various geometric figures. For example, in this exercise, we find the center of a circle using the intersecting properties with a parabola at a specific point. This requires a deep understanding of both the equations governing these figures and the algebraic manipulations necessary to derive solutions.

Tangent to a Curve

A tangent is a straight line that just touches a curve at a point without crossing it. The tangent to a curve indicates the immediate direction in which the curve heads at that point. Understanding the concept of tangents is crucial in many geometry problems, particularly where curves and lines interact. The slope of the tangent to a curve defined by \( y = f(x) \) at a point \( (x_0, y_0) \) is given by the derivative \( f'(x_0) \). In our exercise, at the point \((2,4)\), the slope of the parabola \( y = x^2 \) was calculated as 4 using this derivative technique. Similarly, when a circle touches another curve, like a parabola, the tangent's slope on one should be perpendicular to the radius at the point of tangency. This is shown in the condition: \[ m_{circle} \times m_{parabola} = -1 \]This relationship aids in finding unknown parameters like the center of the circle by involving these geometric constraints, thus solving system-based equations to find solutions.