Problem 129 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

Given : A circle, \(2 x^{2}+2 y^{2}=5\) and a parabola, \(y^{2}=4 \sqrt{5} x\). Statement-1 : An equation of a common tangent to these curves is \(y=x+\sqrt{5}\). Statement- \(2:\) If the line, \(y=m x+\frac{\sqrt{5}}{m}(m \neq 0)\) is their common tangent, then \(m\) satisfies $m^{4}-3 m^{2}+2=0 . (a) Statement- 1 is true; Statement- 2 is true; Statement-2 is a correct explanation for Statement-1. (b) Statement- 1 is true; Statement- 2 is true; Statement-2 is not a correct explanation for Statement- 1 . (c) Statement- 1 is true; Statement- 2 is false. (d) Statement- 1 is false; Statement- 2 is true.

Step-by-Step Solution

Answer

(b) Statement-1 and Statement-2 are true, but Statement-2 is not a correct explanation for Statement-1.

Step 1: Convert the Equation of the Circle

The equation of the circle is given as \(2x^2 + 2y^2 = 5\). To simplify, divide the entire equation by 2: \(x^2 + y^2 = \frac{5}{2}\). This is the equation of a circle centered at the origin with radius \(\sqrt{\frac{5}{2}}\).

Step 2: Equation of the Tangent to the Circle

The general equation of a tangent to a circle \(x^2 + y^2 = r^2\) is \(y = mx \pm \sqrt{r^2(1 + m^2)}\). For our circle, \(r = \sqrt{\frac{5}{2}}\), so substituting this in gives: \(y = mx \pm \sqrt{\frac{5}{2}(1 + m^2)}\).

Step 3: Equation of the Tangent to the Parabola

The general equation for a tangent to a parabola \(y^2 = 4ax\) is \(y = mx + \frac{a}{m}\). For our parabola, \(a = \sqrt{5}\), so the tangent equation is \(y = mx + \frac{\sqrt{5}}{m}\).

Step 4: Equate the two tangents

Since both the circle and the parabola have a common tangent, equate the two equations from steps 2 and 3: \(mx \pm \sqrt{\frac{5}{2}(1 + m^2)} = mx + \frac{\sqrt{5}}{m}\).

Step 5: Evaluate m for a Common Tangent

From the previous step, set \(\sqrt{\frac{5}{2}(1 + m^2)} = \frac{\sqrt{5}}{m}\). Square both sides to solve for \(m\): \[\frac{5}{2}(1 + m^2) = \frac{5}{m^2}\] Simplifying gives: \(5m^4 - 3m^2 + 2 = 0\).

Step 6: Analyze Statement-1

The equation of the tangent given in Statement-1 is \(y = x + \sqrt{5}\), which implies \(m = 1\). Substitute \(m = 1\) into the polynomial derived in Step 5 to check validity: \(1^4 - 3(1)^2 + 2 = 0\). This simplifies to 0, showing Statement-1 is correct.

Step 7: Analyze Statement-2

According to Step 5, \(y = mx + \frac{\sqrt{5}}{m}\) truly represents the tangents when \(m\) satisfies \(m^4 - 3m^2 + 2 = 0\). This validates Statement-2. However, both statements are correct independently.

Step 8: Conclusion about Statements

While Statement-1 is true, it is not directly a result of Statement-2, suggesting Statement-2 is not a correct explanation for Statement-1.

Key Concepts

Circle Equation

A circle is a geometric shape that is perfectly round and is defined by its center and radius. The equation of a circle provides a mathematical description of this shape. In its standard form, a circle's equation centered at the origin is given by:

• \( x^2 + y^2 = r^2 \), where \( r \) is the radius.

In the exercise, the given equation is \(2x^2 + 2y^2 = 5\). To convert it to the standard form, we divide the entire equation by 2, yielding \(x^2 + y^2 = \frac{5}{2}\).
This tells us that the circle is centered at the origin with a radius \(\sqrt{\frac{5}{2}}\). The radius \(r\) is always non-negative and represents the distance from any point on the circle to the center.
Understanding this concept is key when solving related problems as it sets the stage for finding tangents or other intersecting shapes relative to the circle.

Parabola Equation

A parabola is a U-shaped curve that can open upwards, downwards, or sideways, depending on its orientation. Its equation is typically expressed in the form \(y^2 = 4ax\) or \(x^2 = 4ay\). These equations define the locus of points that satisfy the quadratic relation.
For a parabola with a vertex at the origin, in the form \(y^2 = 4ax\), the parameter \(a\) represents the distance from the vertex to the focus. In our exercise, the equation of the parabola is given by:

• \(y^2 = 4\sqrt{5}x\)

Here, \(a = \sqrt{5}\), indicating the parabola opens to the right.
When dealing with tangent lines, an important aspect is the formula for the tangent to a parabola: \(y = mx + \frac{a}{m}\), allowing us to find lines that just touch the curve without crossing.
This concept is essential when analyzing relationships between parabolas and other conic sections, such as circles.

Tangent Line

The concept of a tangent line deals with a straight line that touches a curve at exactly one point, without intersecting it at that point. It gives us an instantaneous direction of the curve. In the context of conic sections like circles and parabolas, the tangent line has special characteristics.
For a circle, the tangent line to a circle \(x^2 + y^2 = r^2\) at a point \((x_1, y_1)\) is perpendicular to the radius at the point of contact. The general form of a tangent's equation to the circle is:

• \(y = mx \pm \sqrt{r^2(1 + m^2)}\), where \(m\) is the slope of the line.

Similarly, for a parabola given by \(y^2 = 4ax\), the tangent line is:

• \(y = mx + \frac{a}{m}\)

To find a common tangent, if any, between a circle and a parabola, it usually involves equating these tangent equations and solving for the unknowns.
Understanding this intersection concept helps in identifying the exact points of tangency and solving complex problems involving multiple curved surfaces.