Advertisements
Advertisements
प्रश्न
Show that the curves \[\frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 \text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1\] intersect at right angles ?
Advertisements
उत्तर
\[\text { We have, } \frac{x^2}{a^2 + \lambda_1} + \frac{y^2}{b^2 + \lambda_1} = 1 . . . \left( 1 \right)\]
\[\text { and } \frac{x^2}{a^2 + \lambda_2} + \frac{y^2}{b^2 + \lambda_2} = 1 . . . \left( 2 \right)\]
\[\text { Now we can find the slope of both the curve by differentiating w . r . t x }\]
\[ \Rightarrow \frac{2x}{a^2 + \lambda_1} + \frac{2y\frac{dy}{dx}}{b^2 + \lambda_1} = 0 \text { and } \frac{2x}{a^2 + \lambda_2} + \frac{2y\frac{dy}{dx}}{b^2 + \lambda_2} = 0\]
\[ \Rightarrow \frac{dy}{dx} = - \frac{x}{y} \times \frac{b^2 + \lambda_1}{a^2 + \lambda_1} \text { and } \frac{dy}{dx} = - \frac{x}{y} \times \frac{b^2 + \lambda_2}{a^2 + \lambda_2}\]
\[ \Rightarrow m_1 = - \frac{x}{y} \times \frac{b^2 + \lambda_1}{a^2 + \lambda_1} \text { and } m_2 = - \frac{x}{y} \times \frac{b^2 + \lambda_2}{a^2 + \lambda_2}\]
\[\text { Subtracting} \left( 2 \right) \text { from } \left( 1 \right), \text { we get }, \]
\[ x^2 \left( \frac{1}{a^2 + \lambda_1} - \frac{1}{a^2 + \lambda_2} \right) + y^2 \left( \frac{1}{b^2 + \lambda_1} - \frac{1}{b^2 + \lambda_2} \right) = 0\]
\[ \Rightarrow \frac{x^2}{y^2} = \frac{\lambda_2 - \lambda_1}{\left( b^2 + \lambda_1 \right)\left( b^2 + \lambda_2 \right)} \times \frac{1}{\frac{\lambda_1 - \lambda_2}{\left( a^2 + \lambda_1 \right)\left( a^2 + \lambda_2 \right)}}\]
\[\text { Now,} \]
\[ m_1 \times \times m_2 = \frac{x^2}{y^2} \times \frac{b^2 + \lambda_1}{a^2 + \lambda_1} \times \frac{b^2 + \lambda_2}{a^2 + \lambda_2}\]
\[ = \frac{\lambda_2 - \lambda_1}{\left( b^2 + \lambda_1 \right)\left( b^2 + \lambda_2 \right)} \times \frac{\left( a^2 + \lambda_1 \right)\left( a^2 + \lambda_2 \right)}{\lambda_1 - \lambda_2} \times \frac{b^2 + \lambda_1}{a^2 + \lambda_1} \times \frac{b^2 + \lambda_2}{a^2 + \lambda_2}\]
\[ = - 1\]
\[\text { hence,} \left( 1 \right) \text { and } \left( 2 \right) \text { cuts orthogonally } . \]
APPEARS IN
संबंधित प्रश्न
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Show that the equation of normal at any point t on the curve x = 3 cos t – cos3t and y = 3 sin t – sin3t is 4 (y cos3t – sin3t) = 3 sin 4t
Find the equations of the tangent and normal to the curve `x^2/a^2−y^2/b^2=1` at the point `(sqrt2a,b)` .
Find the slope of the tangent to the curve y = x3 − 3x + 2 at the point whose x-coordinate is 3.
Find the slope of the normal to the curve x = acos3θ, y = asin3θ at `theta = pi/4`
Find a point on the curve y = (x − 2)2 at which the tangent is parallel to the chord joining the points (2, 0) and (4, 4).
The slope of the tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2,– 1) is
(A) `22/7`
(B) `6/7`
(C) `7/6`
(D) `(-6)/7`
Find a point on the curve y = x3 − 3x where the tangent is parallel to the chord joining (1, −2) and (2, 2) ?
Find the points on the curve\[\frac{x^2}{4} + \frac{y^2}{25} = 1\] at which the tangent is parallel to the y-axis ?
Find the equation of the tangent and the normal to the following curve at the indicated point y = x4 − 6x3 + 13x2 − 10x + 5 at x = 1?
Find the equation of the tangent and the normal to the following curve at the indicated point \[y^2 = \frac{x^3}{4 - x}at \left( 2, - 2 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( x_0 , y_0 \right)\] ?
Find the equation of the tangent and the normal to the following curve at the indicated point 4x2 + 9y2 = 36 at (3cosθ, 2sinθ) ?
Find the equation of the tangent and the normal to the following curve at the indicated point \[\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \text { at } \left( \sqrt{2}a, b \right)\] ?
Find the equation of the normal to the curve ay2 = x3 at the point (am2, am3) ?
Find the equations of all lines having slope 2 and that are tangent to the curve \[y = \frac{1}{x - 3}, x \neq 3\] ?
Find the equation of the tangent to the curve x2 + 3y − 3 = 0, which is parallel to the line y= 4x − 5 ?
Find the angle of intersection of the following curve y = x2 and x2 + y2 = 20 ?
Find the angle of intersection of the following curve \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\] and x2 + y2 = ab ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Write the coordinates of the point on the curve y2 = x where the tangent line makes an angle \[\frac{\pi}{4}\] with x-axis ?
Write the angle made by the tangent to the curve x = et cos t, y = et sin t at \[t = \frac{\pi}{4}\] with the x-axis ?
Write the angle between the curves y2 = 4x and x2 = 2y − 3 at the point (1, 2) ?
Write the equation of the tangent drawn to the curve \[y = \sin x\] at the point (0,0) ?
The point on the curve y2 = x where tangent makes 45° angle with x-axis is ______________ .
The equation of the normal to the curve 3x2 − y2 = 8 which is parallel to x + 3y = 8 is ____________ .
At what point the slope of the tangent to the curve x2 + y2 − 2x − 3 = 0 is zero
If the curve ay + x2 = 7 and x3 = y cut orthogonally at (1, 1), then a is equal to _____________ .
The angle of intersection of the curves y = 2 sin2 x and y = cos 2 x at \[x = \frac{\pi}{6}\] is ____________ .
Show that the equation of normal at any point on the curve x = 3cos θ – cos3θ, y = 3sinθ – sin3θ is 4 (y cos3θ – x sin3θ) = 3 sin 4θ
The equation of normal to the curve 3x2 – y2 = 8 which is parallel to the line x + 3y = 8 is ______.
The tangent to the curve y = 2x2 - x + 1 is parallel to the line y = 3x + 9 at the point ____________.
The tangent to the curve y = x2 + 3x will pass through the point (0, -9) if it is drawn at the point ____________.
The line y = x + 1 is a tangent to the curve y2 = 4x at the point
Tangent is drawn to the ellipse `x^2/27 + y^2 = 1` at the point `(3sqrt(3) cos theta, sin theta), 0 < 0 < 1`. The sum of the intercepts on the axes made by the tangent is minimum if 0 is equal to
The Slope of the normal to the curve `y = 2x^2 + 3 sin x` at `x` = 0 is
