Key Concepts and Formulas Used

To effectively solve problems involving tangents to conic sections, a strong grasp of the following fundamental concepts is essential. These principles allow us to translate geometric conditions into algebraic equations and solve for unknown parameters.

1. Standard Form of an Ellipse and Tangent Equation: For an ellipse given by the general equation $Ax^2 + By^2 = C$, if a point $(x_1, y_1)$ lies on the ellipse, then the equation of the tangent line to the ellipse at that specific point is given by: $Axx_1 + Byy_1 = C$

   • Why this is important: This formula is a direct and efficient way to find the equation of a tangent without needing to differentiate implicitly every time. It's derived from the concept of polar lines and is a standard result in coordinate geometry.

2. Slope of a Line from its Equation: Given a linear equation in the standard form $Px + Qy + R = 0$, the slope $m$ of the line can be readily calculated as: $m = -\frac{P}{Q}$ Alternatively, by rearranging the equation into the slope-intercept form $y = mx + c$, the coefficient of $x$ directly gives the slope $m$.

   • Why this is important: The slope is a numerical measure of a line's steepness and direction. It's crucial for applying conditions like perpendicularity.

3. Condition for Perpendicular Lines: If two lines have slopes $m_1$ and $m_2$ respectively, and they are perpendicular to each other (i.e., they intersect at a 90-degree angle), then the product of their slopes must be equal to $-1$: $m_1 m_2 = -1$

   • Why this is important: This is the core geometric condition given in the problem statement. It allows us to establish a relationship between the slopes of the two tangents.

4. Point Lying on a Curve: If a point $(x_1, y_1)$ lies on a curve defined by an equation, then substituting its coordinates into the curve's equation must satisfy the equation. That is, the equation holds true for $x=x_1$ and $y=y_1$.

   • Why this is important: This fundamental property allows us to create an algebraic equation involving the coordinates of a point if we know it lies on a particular curve. In this problem, it provides a second crucial equation to solve for the unknown coordinates.

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Step-by-Step Solution

Let's systematically approach this problem, breaking it down into manageable steps and explaining the reasoning behind each action.

1. Understand and Identify the Given Information

First, we need to clearly identify the equation of the ellipse and the points of tangency provided in the problem.

• The equation of the ellipse is: $4x^2 + y^2 = 8$ Comparing this to the general form $Ax^2 + By^2 = C$, we have $A=4$, $B=1$, and $C=8$.

• We are given two points on this ellipse where tangents are drawn:

  • Point 1: $P_1(x_1, y_1) = (1, 2)$
  • Point 2: $P_2(x_2, y_2) = (a, b)$

  Why this step? Clearly stating the given information is the foundational step. It helps us identify the specific parameters of the ellipse and the points involved, which are essential for correctly applying the tangent formula and other concepts.

2. Determine the Equation and Slope of the First Tangent (at $P_1(1, 2)$)

We will find the equation of the tangent at point $P_1(1, 2)$ using the tangent formula $Axx_1 + Byy_1 = C$. Substitute $A=4$, $B=1$, $C=8$, and the coordinates of $P_1$, $x_1=1$ and $y_1=2$, into the formula: $4x(1) + 1y(2) = 8$ $4x + 2y = 8$ To simplify this equation, we can divide all terms by 2: $2x + y = 4$ Now, we need to find the slope of this tangent, let's call it $m_1$. We can rearrange the equation into the slope-intercept form ($y = mx + c$): $y = -2x + 4$ By comparing this with $y = mx + c$, we can directly identify the slope of the first tangent: $m_1 = -2$

3. Determine the Equation and Slope of the Second Tangent (at $P_2(a, b)$)

Next, we find the equation of the tangent at the second point $P_2(a, b)$, again using the tangent formula $Axx_1 + Byy_1 = C$. Substitute $A=4$, $B=1$, $C=8$, and the coordinates of $P_2$, $x_1=a$ and $y_1=b$, into the formula: $4x(a) + 1y(b) = 8$ $4ax + by = 8$ Now, let's find the slope of this tangent, $m_2$. Using the formula $m = -P/Q$ for a line $Px + Qy + R = 0$, where here $P=4a$, $Q=b$, and $R=-8$: $m_2 = -\frac{4a}{b}$

4. Apply the Perpendicularity Condition

We are given that the two tangents are perpendicular to each other. According to the condition for perpendicular lines, the product of their slopes must be $-1$: $m_1 m_2 = -1$ Substitute the values of $m_1 = -2$ (from Step 2) and $m_2 = -\frac{4a}{b}$ (from Step 3) into this condition: $(-2) \left(-\frac{4a}{b}\right) = -1$ Simplify the left side of the equation: $\frac{8a}{b} = -1$ Now, we can establish a crucial relationship between $a$ and $b$ by cross-multiplying: $8a = -b$ $b = -8a$ This equation provides a critical link between the coordinates $a$ and $b$ of the second point of tangency.

5. Utilize the Fact that $(a, b)$ Lies on the Ellipse

The point $(a, b)$ is not just an arbitrary point; it is a point on the ellipse $4x^2 + y^2 = 8$. This means its coordinates must satisfy the ellipse's equation. Substitute $x=a$ and $y=b$ into the ellipse equation: $4a^2 + b^2 = 8$ Now, we have a system of two algebraic equations with two unknowns ($a$ and $b$):

1. $b = -8a$ (derived from the perpendicularity condition in Step 4)

2. $4a^2 + b^2 = 8$ (derived from the point $(a,b)$ lying on the ellipse)

   Why this step? The perpendicularity condition alone (equation 1) gives us a relationship between $a$ and $b$, but it doesn't allow us to find a unique numerical value for $a^2$ or $b^2$. The fact that $(a, b)$ is a point on the ellipse provides a second, independent equation, forming a solvable system. This is a common strategy in coordinate geometry problems: combine geometric conditions with the equations of the curves themselves.

6. Solve the System of Equations for $a^2$

Substitute the expression for $b$ from equation (1) into equation (2): $4a^2 + (-8a)^2 = 8$ Carefully square the term $(-8a)$: $(-8a)^2 = (-8)^2 a^2 = 64a^2$. $4a^2 + 64a^2 = 8$ Combine the like terms on the left side: $68a^2 = 8$ Finally, solve for $a^2$: $a^2 = \frac{8}{68}$ To simplify the fraction, divide both the numerator and the denominator by their greatest common divisor, which is 4: $a^2 = \frac{8 \div 4}{68 \div 4}$ $a^2 = \frac{2}{17}$

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Tips and Common Mistakes to Avoid

• Tangent Formula Precision: Always ensure you use the correct tangent formula for the given form of the conic section. For $Ax^2 + By^2 = C$, it's $Axx_1 + Byy_1 = C$. If the ellipse were in the form $\frac{x^2}{P^2} + \frac{y^2}{Q^2} = 1$, the tangent would be $\frac{xx_1}{P^2} + \frac{yy_1}{Q^2} = 1$. While mathematically equivalent, using the most direct form for the given equation minimizes potential errors.
• Slope Calculation Accuracy: Be extremely careful with algebraic signs when calculating the slope from a linear equation. A common error is to forget the negative sign in $m = -P/Q$ or to make a mistake when rearranging into $y=mx+c$.
• Perpendicular vs. Parallel: Do not confuse the condition for perpendicular lines ($m_1 m_2 = -1$) with that for parallel lines ($m_1 = m_2$). These are distinct geometric conditions.
• Utilize All Information: Always make sure you've used every piece of information provided in the problem statement. In this case, both the perpendicularity of tangents and the fact that $(a, b)$ lies on the ellipse were absolutely crucial for finding a unique solution. Missing one would lead to an unsolvable or indeterminate system.
• Algebraic Precision: Double-check all algebraic manipulations, especially substitutions, squaring negative terms (e.g., $(-8a)^2 = 64a^2$, not $-64a^2$), and fraction simplifications. Small errors can easily lead to an incorrect final answer.

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Summary and Key Takeaway

This problem is an excellent illustration of how to combine geometric conditions with algebraic properties in coordinate geometry. The solution path involved:

1. Formulating tangent equations at two distinct points on the ellipse.
2. Calculating the slopes of these tangent lines.
3. Applying the geometric condition of perpendicularity to establish a fundamental algebraic relationship ($b = -8a$) between the coordinates of the unknown point $(a, b)$.
4. Leveraging the algebraic condition that the point $(a, b)$ must lie on the ellipse to form a second independent equation ($4a^2 + b^2 = 8$).
5. Solving the resulting system of two equations to find the required value $a^2$.

The key takeaway is the systematic approach: break down the problem into smaller, manageable parts, apply relevant formulas and conditions step-by-step, and effectively combine different pieces of information (geometric relationships and algebraic curve definitions) to arrive at the solution. This methodical process is vital for success in JEE Mathematics.

The final answer is $a^2 = \frac