Advertisements
Advertisements
प्रश्न
At what points on the curve y = x2 − 4x + 5 is the tangent perpendicular to the line 2y + x = 7?
Advertisements
उत्तर
Let (x1, y1) be the required point.
Slope of the given line = \[\frac{- 1}{2}\]
Slope of the line perpendicular to this line = 2
\[\text { Since, the point lies on the curve } . \]
\[\text { Hence }, y_1 = {x_1}^2 - 4 x_1 + 5 . . . \left( 1 \right)\]
\[\text { Now }, y = x^2 - 4x + 5 \]
\[ \therefore \frac{dy}{dx} = 2x - 4\]
\[\text { Now }, \]
\[\text { Slope of the tangent at }\left( x_1 , y_1 \right)= \left( \frac{dy}{dx} \right)_\left( x_1 , y_1 \right) =2 x_1 -4\]
\[\text { Slope of the tangent at }\left( x_1 , y_1 \right)=\text { Slope of the given line [Given]}\]
\[ \therefore 2 x_1 - 4 = 2\]
\[ \Rightarrow 2 x_1 = 6\]
\[ \Rightarrow x_1 = 3\]
\[\text {Also }, \]
\[ y_1 = 9 - 12 + 5 = 2 [\text { From eq. } (1)]\]
\[\text { Thus, the required point is }\left( 3, 2 \right).\]
APPEARS IN
संबंधित प्रश्न
Find the slope of the tangent to the curve y = (x -1)/(x - 2), x != 2 at x = 10.
Find the slope of the tangent to curve y = x3 − x + 1 at the point whose x-coordinate is 2.
Find points at which the tangent to the curve y = x3 − 3x2 − 9x + 7 is parallel to the x-axis.
Find the equations of the tangent and normal to the parabola y2 = 4ax at the point (at2, 2at).
The line y = mx + 1 is a tangent to the curve y2 = 4x if the value of m is
(A) 1
(B) 2
(C) 3
(D) 1/2
Find the slope of the tangent and the normal to the following curve at the indicted point \[y = \sqrt{x} \text { at }x = 9\] ?
Find the slope of the tangent and the normal to the following curve at the indicted point y = (sin 2x + cot x + 2)2 at x = π/2 ?
Find the values of a and b if the slope of the tangent to the curve xy + ax + by = 2 at (1, 1) is 2 ?
Find the point on the curve y = x2 where the slope of the tangent is equal to the x-coordinate of the point ?
Find the point on the curve y = 3x2 + 4 at which the tangent is perpendicular to the line whose slop is \[- \frac{1}{6}\] ?
Find the points on the curve y = x3 where the slope of the tangent is equal to the x-coordinate of the point ?
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 tangent line to the curve y = x2 + 4x − 16 which is parallel to the line 3x − y + 1 = 0 ?
Prove that \[\left( \frac{x}{a} \right)^n + \left( \frac{y}{b} \right)^n = 2\] touches the straight line \[\frac{x}{a} + \frac{y}{b} = 2\] for all n ∈ N, at the point (a, b) ?
Find the equation of the tangents to the curve 3x2 – y2 = 8, which passes through the point (4/3, 0) ?
Find the angle of intersection of the following curve x2 + 4y2 = 8 and x2 − 2y2 = 2 ?
Find the angle of intersection of the following curve x2 = 27y and y2 = 8x ?
Show that the following curve intersect orthogonally at the indicated point x2 = y and x3 + 6y = 7 at (1, 1) ?
Show that the curves 4x = y2 and 4xy = k cut at right angles, if k2 = 512 ?
Find the condition for the following set of curve to intersect orthogonally \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \text { and } \frac{x^2}{A^2} - \frac{y^2}{B^2} = 1\] ?
Find the point on the curve y = x2 − 2x + 3, where the tangent is parallel to x-axis ?
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
The normal to the curve x2 = 4y passing through (1, 2) is _____________ .
Find the equation of all the tangents to the curve y = cos(x + y), –2π ≤ x ≤ 2π, that are parallel to the line x + 2y = 0.
Prove that the curves xy = 4 and x2 + y2 = 8 touch each other.
The curve y = `x^(1/5)` has at (0, 0) ______.
The tangent to the curve y = e2x at the point (0, 1) meets x-axis at ______.
The slope of tangent to the curve x = t2 + 3t – 8, y = 2t2 – 2t – 5 at the point (2, –1) is ______.
The point on the curves y = (x – 3)2 where the tangent is parallel to the chord joining (3, 0) and (4, 1) is ____________.
The tangent to the curve y = x2 + 3x will pass through the point (0, -9) if it is drawn at the point ____________.
Tangents to the curve x2 + y2 = 2 at the points (1, 1) and (-1, 1) are ____________.
If `tan^-1x + tan^-1y + tan^-1z = pi/2`, then
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 points at which the tangent passes through the origin for the curve y = 4x3 – 2x5 are
Let `y = f(x)` be the equation of the curve, then equation of normal is
If (a, b), (c, d) are points on the curve 9y2 = x3 where the normal makes equal intercepts on the axes, then the value of a + b + c + d is ______.
If m be the slope of a tangent to the curve e2y = 1 + 4x2, then ______.
If the tangent to the conic, y – 6 = x2 at (2, 10) touches the circle, x2 + y2 + 8x – 2y = k (for some fixed k) at a point (α, β); then (α, β) is ______.
