Let a tangent to the Curve 9x² + 16y² = 144 intersect the coordinate axes at the points A and B. The

Q: Why can't we just use the derivative to find the minimum?

You absolutely can! Calculus is a powerful tool. However, the AM-GM inequality is often faster and more elegant for expressions involving sums of reciprocals. It highlights the symmetry of the ellipse.

Q: Why do we use parametric coordinates? Can't we use $y = mx + c$?

You can, but the condition for tangency $c^2 = a^2m^2 + b^2$ can get messy. Parametric coordinates simplify the tangent equation significantly, making the minimization step much cleaner.

Q: Is the minimum length always $a+b$?

For this specific ellipse, yes! The minimum length of the tangent segment intercepted by the axes is $a+b$. It's a beautiful geometric property of the ellipse.

Question

Let a tangent to the Curve $9x^2 + 16y^2 = 144$ intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is _____.

Sigma Percentile · Accepted Answer

### Analyzing the Geometry of the Ellipse

Welcome, future engineer. Today, we are uncovering the hidden geometry of the ellipse defined by the equation $9x^2 + 16y^2 = 144$.

To standardize this, we divide the entire equation by $144$ to reveal the canonical form:

$$
\frac{x^2}{16} + \frac{y^2}{9} = 1
$$

This is the standard equation of an ellipse where $a^2 = 16$ and $b^2 = 9$. Thus, our semi-major axis is $a = 4$ and our semi-minor axis is $b = 3$.

### The Parametric Dance

Imagine a point $P$ moving along this ellipse. Instead of dealing with Cartesian coordinates, we utilize the parametric form:

$$
P(4\cos	heta, 3\sin	heta)
$$

This single parameter, $	heta$, allows us to describe any point on the curve with elegance. When we draw a tangent at this point using the standard formula $\frac{xx_1}{a^2} + \frac{yy_1}{b^2} = 1$, we substitute our point $P$ to obtain:

$$
\frac{x(4\cos	heta)}{16} + \frac{y(3\sin	heta)}{9} = 1
$$

This simplifies beautifully to the tangent equation:

$$
\frac{x\cos	heta}{4} + \frac{y\sin	heta}{3} = 1
$$

### The Tangent's Reach

This tangent line acts as a bridge between the coordinate axes. To find the x-intercept, we set $y=0$, yielding $x = 4\sec	heta$.

Setting $x=0$ gives the y-intercept, $y = 3\csc	heta$. These define our intersection points $A(4\sec	heta, 0)$ and $B(0, 3\csc	heta)$.

We aim to minimize the length $L$ of the segment $AB$. Using the distance formula, we express $L^2$ as:

$$
L^2 = (4\sec	heta)^2 + (3\csc	heta)^2 = 16\sec^2	heta + 9\csc^2	heta
$$

### The AM-GM Masterpiece

We now minimize $L^2 = 16\sec^2	heta + 9\csc^2	heta$ by applying the trigonometric identities $\sec^2	heta = 1 + 	an^2	heta$ and $\csc^2	heta = 1 + \cot^2	heta$. Substituting these, we get:

$$
L^2 = 16(1 + 	an^2	heta) + 9(1 + \cot^2	heta) = 25 + 16	an^2	heta + 9\cot^2	heta
$$

Here, the **AM-GM inequality** shines. For positive numbers, the sum is minimized when the terms are equal. The variable part $16	an^2	heta + 9\cot^2	heta$ has a minimum value of:

$$
2\sqrt{(16	an^2	heta)(9\cot^2	heta)} = 2\sqrt{144} = 24
$$

Thus, $L^2 \ge 25 + 24 = 49$. Taking the square root, we find the **minimum length $L_{min} = 7$**.

The elegance of this result is breathtaking, as the minimum length is simply $a+b$. You have mastered the geometry, the algebra, and the optimization.

Animated Solution for Mathematics - Conic Sections: Let a tangent to the Curve $9 x^{2} + 16 y^{2} = 144$ intersect the coordinate axes at the points A and B. Then, the minimum length of the line segment AB is _____.

Enter Numerical Value:

Visualized Solution

Standardizing the Equation

Extracting Ellipse Parameters

Defining the Parametric Point $P$

Equation of the Tangent Line

Finding the Intercepts $A$ and $B$

Setting up the Length $A B$

Trigonometric Transformation

Applying AM-GM Inequality

Evaluating the Minimum Value

Final Minimum Length

The Sigma Insight: Equation of Tangent and Normal

Analyzing the Geometry of the Ellipse

The Parametric Dance

The Tangent's Reach

The AM-GM Masterpiece

Similar Questions

Let $A B$ be a chord of length 12 of the circle $(x - 2)^{2} + (y + 1)^{2} = \frac{169}{4}$ . If tangents drawn to the circle at points $A$ and $B$ intersect at the point $P$ , then five times the distance of point $P$ from chord $A B$ is equal to ________

The minimum area of triangle formed by the tangent to the $\frac{x ^{2}}{a ^{2}} + \frac{y ^{2}}{b ^{2}} = 1$ & coordinate axes is

Let a line perpendicular to the line $2 x - y = 10$ touch the parabola $y^{2} = 4 (x - 9)$ at the point $P$ . The distance of the point $P$ from the centre of the circle $x^{2} + y^{2} - 14 x - 8 y + 56 = 0$ is __________

Let $P$ be the point on the parabola $y^{2} = 4 x$ which is at the shortest distance from the center $S$ of the circle $x^{2} + y^{2} - 4 x - 16 y + 64 = 0$ . Let $Q$ be the point on the circle dividing the line segment $S P$ internally. Then

Let the tangent to the parabola $y^{2} = 12 x$ at the point $(3, α)$ be perpendicular to the line $2 x + 2 y = 3$ . Then the square of distance of the point $(6, - 4)$ from the normal to the hyperbola $α^{2} x^{2} - 9 y^{2} = 9 α^{2}$ at its point $(α - 1, α + 2)$ is equal to .............

Let the lengths of intercepts on $x$ -axis and $y$ -axis made by the circle $x^{2} + y^{2} + a x + 2 a y + c = 0$ ( $a < 0$ ) be $22$ and $25$ , respectively. Then the shortest distance from origin to a tangent to this circle which is perpendicular to the line $x + 2 y = 0$ , is equal to :

The shortest distance between the curves $y^{2} = 8 x$ and $x^{2} + y^{2} + 12 y + 35 = 0$ is :

Let $2 x^{2} + y^{2} - 3 x y = 0$ be the equation of a pair of tangents drawn from the origin $O$ to a circle of radius 3 with centre in the first quadrant. If $A$ is one of the points of contact, find the length of $O A$ .