The two tangents to the curve a x^2+2 h x y+b y^2=1, a>0 at the points where it crosses x-axis,

The two tangents to the curve a x^2+2 h x y+b y^2=1, a>0 at...

Key Concepts

Parallel Lines

Parallel lines are lines in a plane that never intersect, characterized by having the same slope. After calculating the slopes of the tangents at the points where the conic crosses the x-axis, the result shows that they are identical. This identical slope indicates that the two tangent lines, though drawn at different points, are parallel to each other.

Implicit Differentiation

Implicit differentiation is a technique used to differentiate equations that define y implicitly as a function of x. This method is essential when dealing with conic sections, as it allows for the determination of the slope of the tangent line at a given point on the curve. By applying implicit differentiation to the conic’s equation, the slope of the tangent at the x-axis intercepts is obtained.

X-Axis Intercepts

Finding the x-axis intercepts involves setting y equal to zero in the equation of the conic. This procedure simplifies the equation, making it easier to solve for the points where the curve crosses the x-axis. These intersection points are crucial for identifying the specific locations at which the tangents are drawn.

Conic Sections

Conic sections are curves obtained by intersecting a plane with a double-napped cone. In the given problem, the curve is defined by a general second degree equation in x and y, which describes various conics such as ellipses, hyperbolas, and parabolas depending on the coefficients. Understanding the general formula for conic sections helps in determining their properties, such as intercepts and tangent lines.

Tangent Lines to Conic Sections

A tangent line to a conic section is a line that touches the conic at exactly one point without crossing it at that vicinity. Determining the equation of a tangent often begins with finding the derivative of the conic's equation through implicit differentiation. In this problem, the tangents at the x-axis intersections are studied to understand their slope relationship.

Transcript

00:01 The question given curve is a x square plus 2 h xy plus b y square is equal to 1.

00:09 And in the question it is said that this curve cuts the x -axis.

00:14 That means at x -axis we know y is equal to 0.

00:19 So put this value here and we get a x square plus plus 0 plus 0 that is equal to 1.

00:28 So from here i can write that x equal to plus minus 1 by under root a that means this curl cuts the x -axis at two points one is one by under root 2 1 by under root a and the other point is minus 1 by under root a now if we if we find slope of the tangent on this curve at these two points so first we write the slope of the tangent that means we need to differentiate this equation with respect to with respect to x so first let's suppose this is equation first first differentiate differentiate first with respect to with respect to with respect to x then we get 2 a x plus 2 h x plus 2 h x d y plus 2 h x d y by d x plus 2 h x d x plus 2 b y, dy by d x that is going to be zero.

01:36 So from here i can write that at at point we have a point x coordinate is plus minus 1 by and the root a and the y coordinate is zero.

01:50 We need to find the slope of the tangents at this point.

01:54 So put this value in the above equation in this in this expression then all the terms which is having y, they will become 0 so from here i can write 2 a x 2 a x 2 a x 2a x plus 2 h x d y by d x that is that is going to be 0 that is going to be 0 now put the value of x also 2a x value is plus minus 1 by under root a plus this is only plus here plus 2h value of x is plus minus 1 by and root a multiply d .y by d x that is going to be 0.

02:42 From here i can write that a plus h, d y, by d x that will be equal to 0...

Please give Ace some feedback