(i) Show that the conic 12 x^2+7 x y-10 y^2+13 x+45 y-23=0 is a hyperbola (ii) find the coordina

(i) Show that the conic 12 x^2+7 x y-10 y^2+13 x+45 y-23=0...

Key Concepts

Conic Sections

Conic sections are curves obtained by the intersection of a plane and a double-napped cone, and they can be represented by a general second-degree equation in two variables. Analyzing the quadratic form of this equation helps in identifying the type of conic—ellipse, parabola, or hyperbola—and its geometric properties.

Discriminant Analysis in Conics

The discriminant of a conic section, given by B² - 4AC for an equation Ax² + Bxy + Cy² + ... = 0, is a crucial indicator of the conic's type. If the discriminant is greater than zero, the conic is a hyperbola; if equal to zero, a parabola; and if less than zero, an ellipse. This criterion helps quickly classify the conic without needing to complete other algebraic manipulations.

Centre of a Conic Section

The centre of a conic, particularly for ellipses and hyperbolas, is the point about which the figure is symmetric. It can be found by setting the first partial derivatives of the quadratic form (considering x and y) to zero, or by completing the square in both variables. This centre is critical when translating the conic to a coordinate system centered at the origin for further analysis.

Asymptotes of a Hyperbola

Asymptotes for hyperbolas are the lines that the curve approaches at infinity. They can be determined by setting the higher-degree terms (usually the quadratic part) equal to zero, effectively reducing the conic equation to a pair of linear equations. These lines give insights into the hyperbola's orientation and shape, and are particularly useful when sketching or transforming the hyperbola through rotation or translation.

Transcript

00:01 In this problem of conic section we have given the equation of conic as 12x square plus 7xy minus 10 y square plus 13x plus 45y this is 45y minus 23 is equal to 0 and now we have to show that this is the representing a hyperbola so from here when we consider so a is equal to 12 b is equals to this is b y square so this is minus 10 h is equal so this would be 2 of h so this would be h is equal to 7 divided with 2 and g is equal to this is 2g so this would be 13 divided with 2 and f so this would be 2 f so we can say f is equal to 45 divided with 2 and c is basically the constant term which is minus 23 and now we have to calculate h is square minus so h square is 49 divided with 4 minus ab so this would be minus and minus plus 120 which is greater than 0 and delta so delta is given by a b c plus twice of fg h minus a f is square minus bg square minus c h square and this turn we know that this is not equals to a zero so this is representing a hyper so hence equation is representing a hyperbola.

01:48 So we have proved this term and now for the second part.

01:52 So this is the proof and now for the second part we have this is the second part.

02:00 We have to find the coordinate of its center.

02:04 So center is given why say this is the center in terms of h these terms so this is given why hf so this is hf minus bg divided with ab minus h square so ab minus h square and here this is g h minus a f divided with the same term which is ab minus h square so putting these values so we are getting the term which is minus 575 you can put these values here 575 divided with 500 and 989 divided with 529.

02:50 So this is the center and in third part.

02:54 In third part we have to find the asymptote.

02:56 So asymptote is an asymptote.

02:58 So this is an asymptote.

03:01 So this is approaching towards 0, that is delta is equals to 0...

Please give Ace some feedback