(i) Find focus, vertex, directrix and axes of the conic represented by the equation x^2+2 y-3 x+

(i) Find focus, vertex, directrix and axes of the conic...

Key Concepts

Latus Rectum

The latus rectum is a chord of the conic that passes through the focus and is parallel to the directrix. Its length provides important information about the shape of the conic, including the 'width' of a parabola or details about the ellipse or hyperbola. The length of the latus rectum is directly connected to the parameters found in the standard form of the conic section’s equation.

Eccentricity

Eccentricity is a measure of how much a conic section deviates from being circular. It is defined as the ratio of the distance from any point on the conic to a focus, to its distance to the corresponding directrix. For circles, the eccentricity is zero, for ellipses it lies between zero and one, for parabolas it equals one, and for hyperbolas it is greater than one.

Tangent to Conics

The tangent to a conic section is a straight line that touches the conic at exactly one point. Finding the equation of a tangent often involves using derivative calculus, geometric properties, or the point of tangency condition with the conic’s equation. This process is integral for understanding the behavior of conics at specific points and has applications in optimization and geometric constructions.

Axes of a Conic

The axes of a conic section refer to the lines of symmetry that help define its geometry. For ellipses and hyperbolas, these are the major and minor axes or the transverse and conjugate axes respectively. Identifying these axes is crucial in understanding the orientation, symmetry, and key dimensions of the conic.

Completing the Square

Completing the square is a fundamental algebraic technique used to rewrite quadratic equations into a standard form. This process involves rewriting the quadratic expression as a perfect square plus a constant, which simplifies the identification and analysis of the conic section's key features, such as its vertex or center.

Conic Sections

Conic sections are the curves obtained by intersecting a plane with a right circular cone. The main types include circles, ellipses, parabolas, and hyperbolas. Each type has unique standard forms and geometric properties that allow for the determination of features such as focus, directrix, axes, and eccentricity.

Vertex, Focus, and Directrix

For conic sections, particularly parabolas, the vertex is the point where the curve attains its extreme value, the focus is a fixed point used in the definition of the curve, and the directrix is a fixed line. The distance between any point on the parabola and the focus is equal to its distance from the directrix. This concept extends to ellipses and hyperbolas where the focus and directrix help in defining the shape through eccentricity.

Transcript

00:01 In this problem of conic section we have to find focus vertex directics and axis of conic represented by the equation which is x square plus 2y minus 3x plus 5 is equals to 0.

00:16 So first step is to find the delta so delta here is not equals to 0 so here we can see clearly delta would be not equals to 0.

00:27 Delta is equal to abc so this is not equals to 0.

00:30 And from here we can say this is comparing with the standard equation.

00:35 So this would be a x square or we can say this would be here.

00:41 This is a x square plus b y square.

00:45 So this is b y square plus 2 h xy plus 2g x plus 2g x plus 2 fy plus c is equal to 0.

00:59 And you can see here there is no term of xy xy.

01:01 That means h is equal to 0 and there is no term of y square so here from here we can say this is h square is equals to ab but delta is not equals to 0 so here this would be representing a parabola and also you can see here we have quadratic equation in x and linear in y so this would be a parabola so from here we identified that this is the equation of parabola.

01:35 Now we have to find the focus vertex and directories and axis of coning.

01:40 So here this is equation can be written as say this is this would be x square minus 3x plus 2y plus 5 is equal to 0.

01:55 And now when we make the square so this would be x square minus 3x plus this would be square of this term so this would be 9 divided with this would be 4 because 3 divided with 2 whole square so this is 9 divided with 4 and here this is equals to 2y and what we have added we have to subtract it so this would be minus 11 divided with 4 and now this is the perfect square of x minus 3 divide with 2 whole is square and now this term can be as minus 2 taken common and here we left with y plus this would be here minus also so this would be y plus 11 divided with 8 and now from here we can say vertex so vertex is this would be 3 divided with 2 and here this is minus 11 divided with 8 now we can find focus also so focus would be here 3 divided with 2 and this would be minus d divide with 4a plus 1 divide with 4a so this would be minus 15 divided with 8 and now we can find equation of directics so equation of directives is given by say this would be y plus 11 divided with 8 is equal so this would be here 1 divided with 2 so this is the equation of directrix or when we solve it so this is y is equals to minus 7 divided with 8 and now we can compare it with so you can compare it with the standard form so this is x plus b divided with 2a whole a square is equals to this is 1 divided with a y plus d divided with 4a so here is the equation of directrics so this is equation of directrics and now axis of the conic so axis of the conic is here x minus 3 divided with 2 is equal to 0 that means x is equals to 3 divided with 2 so here axis of conic is x is equals to 3 divided with 2 so this is the right answer vertex focus and the equation of directus and equation of conic and now in part b we have to find the equation of tangent to the parabola so whose equation is y square is equal to 9x which passes through the point say passes through the point 4 .10 and also we have to find the point of contact so here y is equal to 9 x and from here we can say a is equal comparing with the standard form so this would be y is square is equal to 4 a x so a is equals to this would be 9 divided with 4 and now equation of tangent to parabola is y is equals to mx plus a divided with m so since it passes through 4 10 so it must satisfy this point also so putting the value a so a is equals to 9 divided with 4 so this is equals to y is equals to mx plus 9 divided with 4m so this is the equation of tangent and now since it passes to this point so it must satisfy this point so putting this value so this would be y is equals to 10 m is unknown and x is equal to 4 so this would be 4x plus putting here the value so this would be 9 divided with 4m so 9 divided with 4m this would be 4m here also so this is 4m and 4m now when we solve for m so we get m is equals to 1 divided with 4 and 9 divided with 4 so equation of tangent becomes so here we have two values of m so so equation of tangent becomes y is equals to 1 divided with 4x plus 9 divided with 4 and also y is equals to 9 divided with 4x plus 1.

06:55 So this is the equation of tangent and passes through this point.

07:04 And now third problem.

07:06 So here this is third problem.

07:09 So here we have to find the eccentricity, focus, vertex, directics and length of letters rectal.

07:14 And equation of letters act term of the conic.

07:18 So conic is given by x square plus 4 y square plus 2x plus 16 y plus 13 is equal to 0.

07:30 So again from here we can say here we have x square term and y square term and here clearly delta is not equals to 0 and h square is less than ab comparing with the standard form so this would be representing an ellipse so here we are first we have to write it into the form of ellipse so compare it so x square 2x so compare make the perfect square so this can be written as x plus 1 whole square we have added 1 here so this would be 12 and this term can be written as plus 4 and y plus 2 so this is y plus 2 whole square is here this would be 4 and now we have to write it in the form of capital x square divided with two square and here four to four is cancelled out and this would be capital y divided with one is equals to one so this is the standard form capital x is equals to x plus one and capital y is equal to y plus two so now we can find the eccentricity so we know that e square is equals to this would be b square minus a square divided with so this would be b square which is 2 square is 4 minus 1 divided with again 4...

Please give Ace some feedback