The eccentricity of the hyperbola (x)/(a^2)-(y)/(b^2)=1 with latus rectum (32 √(2))/(5) and pass

The eccentricity of the hyperbola (x)/(a^2)-(y)/(b^2)=1 with...

Key Concepts

Hyperbola

A hyperbola is one of the conic sections, defined as the set of all points in a plane such that the absolute difference of their distances from two fixed points (the foci) is constant. It has a distinct geometry characterized by two separate, mirror-image branches with asymptotes that describe its end behavior, and it is usually expressed in a standard form that relates its transverse and conjugate axes.

Eccentricity

The eccentricity of a conic section is a measure of its deviation from being circular. For a hyperbola, the eccentricity (denoted as e) is always greater than 1 and is calculated based on the distance between the center and the foci relative to the lengths of its axes. It helps in specifying the shape of the hyperbola and is a key parameter in many of its geometric properties.

Latus Rectum

The latus rectum of a conic section is a line segment that passes through a focus and is perpendicular to the major axis (or transverse axis for a hyperbola). Its length is directly related to the parameters of the conic and provides important information about the conic’s curvature and width near the focus.

Intersection of Lines

Determining the intersection of lines involves solving a system of linear equations. The point at which these lines meet can be used in various geometrical constructs, such as confirming if a conic section passes through that point or in setting boundary conditions for a problem involving conics.

Transcript

00:01 So in this question, a hyperbola in standard form is given with later set term 32 to 2 by 5 and also given that this hyperbola is passing through the point of intersection of these two lines.

00:15 Since we know that in standard form hyperbola has this equation, the formula for eccentricity is this which is equivalent to saying that b square equals a square times e square minus 1 and also the later section of this hyperbola is equal to 2b.

00:31 Square by a.

00:32 So let's put the value of later setum here 2b square by a and this is given to be 32 root 2 by 5 here.

00:41 So 2 times in place of b square i am writing a square times e square minus 1 using this formula upon a equals 32 root 2 upon 5.

00:55 So after canceling 2 and canceling a i get a times e square minus 1 and as 16 root 2 upon 5 this is our first equation now let's solve these two straight lines first that is 7x plus 13 y equals 87 and 5x minus 8y equals minus 7 so let's multiply 5 in equation 1 i get 35x plus 65 y as 430 and let's multiply 7 in the second equation i get 35x minus 56y as minus 49 now let's subtract these two equations i get this is 121y equals 484 so y equals 4 and if i put y equals 4 here i get 5x equals 8y minus 7 that is 8 times 4 minus 7 32 minus 7 that is 25 so x is 5 so the point of intersection is 5 .4 the hyperbola is set to pass through this point 5 .4 so satisfying 5 form of 4 in the hyperbola's equation i get 25 by a square minus 16 by b square as 1 25 by a square minus 16 upon again i'm applying the formula for b square here this is a square times e square minus 1 as 1 multiplying this equation with a square times e square minus 1 i get 25 times a square minus 1 minus 16 equals a square times a square minus 1 so this is 25 times a square minus 1 so this is 25 times e square minus 25 minus 16 equals a square e square minus a square so this is 25 e square minus 41 equals a square e square minus a square this is our second equation now we have these two equations and we have to solve for a and e here so let's eliminate e square minus 1 here from equation 1 e square minus 1 is equal to 16 root 2 upon 5a let's substitute this value here i get on the right side a square times 16 root 2 upon 5a and on the left side i get 25 into in place of e square i will place 16 root 2 upon 5 a plus 1 and minus 41 right so this is after canceling 5 here i get 5 into 16 root 2 upon a plus 25 minus 41 as on the right side i get 16 root 2 upon 5 times a so taking 16 root 2 common here i get 5 upon a minus a upon 5 from the right side and this equals 16.

05:12 So 16 is getting cancelled and we get 5 upon a minus a upon 5 as 1 upon root 2.

05:29 Now we have to find a from here...

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