Question

If \(2 x-y+1=0\) is a tangent to the hyperbola \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{16}=1\), then which of the following CANNOT be sides of a right angled triangle? [A] \(a, 4,1\) [B] \(a, 4,2\) [C] \(2 a, 8,1\) [D] \(2 a, 4,1\)

Short answer

Option [C] cannot be the sides of a right angled triangle.

Step-by-step solution

Find the Slope of the Tangent

To find the slope of the tangent to the hyperbola, we need to express the equation of the tangent in the slope-intercept form, i.e., y = mx + c. The given equation of the tangent is `2x - y + 1 = 0`. Rearrange it to get `y = 2x + 1`. The slope of the tangent (m) is 2.

Use the Slope of the Tangent to Find the Slope of Normal

For a hyperbola \(\frac{x^2}{a^2}-\frac{y^2}{b^2}=1\), the slopes of tangent and normal at any point are negative reciprocals. Therefore, if the slope of the tangent is 2, the slope of the normal (m') is \(\frac{-1}{2}\).

Apply the Condition for a Right Angled Triangle

The lengths a, b, and c represent the sides of a triangle and, according to the Pythagorean theorem for right triangles, must satisfy \(c^2 = a^2 + b^2\). We compare this with the sides given in options [A] to [D].

Check Each Option

For each set of values in options [A] to [D], check if they satisfy \(c^2 = a^2 + b^2\):- [A]: \(1^2 eq a^2 + 4^2\), doesn't satisfy.- [B]: \(2^2 eq a^2 + 4^2\), doesn't satisfy, but this doesn't prove impossibility without the value of a.- [C]: \(1^2 eq (2a)^2 + 8^2\), doesn't satisfy, but this is also not conclusive.- [D]: \(1^2 eq (2a)^2 + 4^2\), doesn't satisfy, and similarly, this does not provide conclusive proof. Since we don't know the value of a, it's not possible to definitively say which option CANNOT be sides of a right angled triangle without additional information about a.

Determine the Value of a

To eliminate the options, we need to find the value of a. The relation between the hyperbola's a and the slope of the normal can be derived from the equation of the normal to the hyperbola at point \( (a\sec\theta, b\tan\theta) \). For our hyperbola, \(\frac{x^2}{a^2}-\frac{y^2}{16}=1\), we equate the slope of the normal (\frac{-1}{2}) with the slope in the point-slope form \(\frac{b\tan\theta}{a\sec\theta} = \frac{-1}{2}\). Here \(b=4\) (from hyperbola equation). Thus, we find that \(8\tan\theta = -a\sec\theta\). Squaring both sides gives \(64\tan^2\theta = a^2\sec^2\theta\). Converting \(\tan^2\theta\) to \(\sec^2\theta-1\), we have \(64(\sec^2\theta-1) = a^2\sec^2\theta\), which simplifies to \(a^2 = 64\). Therefore, \(a=8\)\. Since we have the value of a, we can re-evaluate options [A] to [D].

Reassess Each Option with the Value of a

Now that we know \(a=8\), we can re-assess the truth of the Pythagorean theorem for each option:- [A]: \(1^2 = 8^2 + 4^2\)? False.- [B]: \(2^2 = 8^2 + 4^2\)? False.- [C]: \(1^2 = (2*8)^2 + 8^2\)? False.- [D]: \(1^2 = (2*8)^2 + 4^2\)? False.However, for a right-angled triangle, the longest side should be the hypotenuse. In [A], [B], and [D], even if we consider \(a=8\), the given sides do not form a right-angled triangle, but they can't be immediately discarded as it is not explicitly stated which side is the hypotenuse. In [C], \(1\) can't be the hypotenuse since the other two sides are longer, so it violates the Pythagorean theorem. Therefore, option [C] cannot be the sides of a right-angled triangle.

Key concepts

Hyperbola Tangent Properties

The study of conic sections such as hyperbolas is fundamental in JEE Advanced Mathematics. A hyperbola is defined by the equation \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\, where \(a\) and \(b\) are the distances from the center to the vertices and co-vertices, respectively. A tangent to a hyperbola is a line that touches the curve at exactly one point, and its equation can significantly aid in analyzing conic properties and solving related problems.

The equation of a tangent to a hyperbola can be derived using the point-slope form if the point of tangency is known, or it can be expressed in terms of its slope if the specific point is not given. Importantly, the angle between the tangent and the axes depends on this slope, with the slope of the tangent at \((a\sec\theta,b\tan\theta)\) being \(b\tan\theta/a\sec\theta\). Knowing these properties is essential for solving questions involving tangents, normals, and even determining characteristics of related geometric figures like triangles.

Slope of Tangent and Normal

In the context of JEE Advanced Mathematics problems, relating the slope of a tangent and normal to the hyperbola is crucial. For the hyperbola \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1\, the slope \( m \) of the tangent line is found by expressing the line in slope-intercept form as \( y = mx + c \). Similarly, the slope \( m' \) of the normal, which is the line perpendicular to the tangent at the same point of contact, is the negative reciprocal of \( m \). Simply put, if \( m = \frac{dy}{dx} \), then \( m' = -\frac{dx}{dy} \).

Understanding the relationship between these slopes is not just a matter of manipulating algebraic expressions; it allows students to navigate through problems involving angles between these lines, finding points of contact, and integrating these concepts with other geometrical shapes such as triangles, which can emerge in complex problems.

Pythagorean Theorem in Triangles

Functionality in JEE Problems

The Pythagorean theorem is a cornerstone of geometry and is incredibly useful in a wide array of mathematical problems, especially in JEE Advanced Mathematics. Stated simply, in a right-angled triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Expressed algebraically, if \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides, then \( c^2 = a^2 + b^2 \).

In the context of the given exercise, the Pythagorean theorem is used to check whether a set of three numbers could represent the sides of a right-angled triangle. This approach to problem-solving is characteristic of many geometry problems where one must deduce possible properties or rules from given information.

Right-Angled Triangle Condition

Determining whether a set of sides can form a right-angled triangle involves an application of the Pythagorean theorem. However, it's not just about plugging values into the theorem's formula; one must also consider the conditions under which a right-angled triangle exists. Specifically, for three lengths to form a right-angled triangle, the longest of the three must serve as the hypotenuse. This aligns with the requirement that in a right-angled triangle, the angle opposite the hypotenuse is 90 degrees.

In JEE mathematics problems, when presented with a set of possible triangle sides, students must evaluate not only the Pythagorean theorem's satisfaction but also the logical placement of sides based on length. As shown in the step by step solution, recognizing that the given side lengths do not align with the hypotenuse condition can be crucial for solving problems and eliminating incorrect answer choices, such as in option [C], where the length '1' simply cannot be the hypotenuse.