23 Maths -- Analytic Geometry

We obtain expression for trigonometric functions by solving for right angle triangle drawn on a circle. Similarly, how are the expressions for the functions of the type ex - e-x^2 , 2ex +e-x , ............ , ex + e-x^2 ? Provide an analytic explanation.

We obtain expression for trigonometric functions by solving for right angle triangle drawn on a circle. Similarly, how are the expressions for the functions of the type e^x - e^{-x^2} , 2e^x +e^-x , ............ , e^x + e^{-x^2} ? Provide an analytic explanation.

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The image of the point (-8,12) with respect to the line mirror 4x+7y+13=0 is (a) (16,-2) (b) (-16,2) (c) (16,2) (d) (-16,-2)

Answer is option d. (-16,-2)

Explanation:-

Show that the lines joining the origin to the point of intersection of the line x + y = 1 with the curve 4x2+4y2+4x-2y-5=0 are at right angles to each other.

If the line pairs ax2+2hxy+by2=0 and a'x2+2h'xy+b'y2=0 have the same bisectors, prove that h(a'-b')=h'(a-b).

Find the equation to the pair of straight lines joining the origin to the intersection of the straight line y=mx+c and the curve x2+y2=a2. Prove that they are at right angles if 2c2=a2(1+m2).

Prove that the two straight lines (x2+y2)sin2α=(xcosθ - ysinθ)2 include an angle 2α.

Find the condition that one of the lines ax2+2hxy+by2=0 may coincide with one of the lines a1x2+2h1xy+b1y2=0.

Prove that the product of the perpendicular drawn from the point (x1,y1) on the lines represented by ax2+2hxy+by2=0 is (ax12+2hx1y1+by12)/(√{(a-b)2+4h2}.

Show that the lines joining the origin to the points common to x2 + hxy - y2+ gx + fy = 0 and fx - gy = λ are at the right angles for all values of λ ≠ 0.

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